STRAIGHT LINES

&

CIRCLES - I

Quest

Select the correct alternative : (Only one is correct)

Q.1 If the lines x + y + 1 = 0 ; 4x + 3y + 4 = 0 and x + αy + β = 0, where α² + β² = 2, are concurrent then (A) α = 1, β = – 1 (B) α = 1, β = ± 1 (C) α = – 1, β = ± 1 (D) α = ± 1, β = 1

Q.2 The axes are translated so that the new equation of the circle x²+y²−5x+2y –5 = 0 has no first degree terms. Then the new equation is :

(A) x² +y² = 9 (B) x² +y² = 49

4 (C) x² +y² = ⁸¹

16 (D) none of these

Q.3 Given the family of lines, a(3x + 4y + 6) + b(x + y + 2) = 0 . The line of the family situated at the greatest distance from the point P (2, 3) has equation :

(A) 4x + 3y + 8 = 0 (B) 5x + 3y + 10 = 0 (C) 15x + 8y + 30 = 0 (D) none

Q.4 The ends of a quadrant of a circle have the coordinates (1, 3) and (3, 1) then the centre of the such a circle is

(A) (1, 1) (B) (2, 2) (C) (2, 6) (D) (4, 4)

Q.5 The straight line, ax + by = 1 makes with the curve px² + 2axy + qy² = r a chord which subtends a right angle at the origin . Then :

(A) r (a² + b²) = p + q (B) r (a² + p²) = q + b (C) r (b² + q²) = p + a (D) none

Q.6 The circle described on the line joining the points (0,1), (a,b) as diameter cuts the x−axis in points whose abscissae are roots of the equation :

(A) x²+ ax + b = 0 (B) x²−ax+ b = 0 (C) x²+ax−b = 0 (D) x²−ax−b = 0

Q.7 Centroid of the triangle, the equations of whose sides are 12x² – 20xy + 7y² = 0 and 2x – 3y + 4=0 is

(A) (3, 3) (B) 



 



 3 ,8 3 8

(C) 



 



 3 ,8

3 (D) 



 



 ,3 3 8

Q.8 The line 2x – y + 1 = 0 is tangent to the circle at the point (2, 5) and the centre of the circles lies on x – 2y = 4. The radius of the circle is

(A) 3 5 (B) 5 3 (C) 2 5 (D) 5 2

Q.9 The line x + 3y − 2 = 0 bisects the angle between a pair of straight lines of which one has equation x − 7y + 5 = 0 . The equation of the other line is :

(A) 3x + 3y − 1 = 0 (B) x − 3y + 2 = 0 (C) 5x + 5y − 3 = 0 (D) none

Q.10 Given two circles x²+ y²− 6x− 2y+ 5 = 0 & x²+ y²+ 6x+ 22y+ 5 = 0. The tangent at (2, −1) to the first circle :

(A) passes outside the second circle (B) touches the second circle

(C) intersects the second circle in 2 real points (D) passes through the centre of the second circle.

Q.11 A variable rectangle PQRS has its sides parallel to fixed directions. Q & S lie respectively on the lines x = a, x = −a & P lies on the x−axis . Then the locus of R is :

(A) a straight line (B) a circle (C) a parabola (D) pair of straight lines Q.12 To which of the following circles, the line y−x+3 = 0 is normal at the point ₃ ³

2 3

2

 +

 



, ?

(A) x− − y

 

 + −

 

 = 3 3

2

3

2 9

2 2

(B) x− y

 

 + −

 

 = 3

2

3

2 9

2 2

(C) x²+ (y−3)² = 9 (D) (x−3)² + y² = 9

[3]

Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Q.13 On the portion of the straight line, x + 2y = 4 intercepted between the axes, a square is constructed on the side of the line away from the origin. Then the point of intersection of its diagonals has co-ordinates

(A) (2, 3) (B) (3, 2) (C) (3, 3) (D) (2, 2)

Q.14 The locus of the mid point of a chord of the circle x²+y² = 4 which subtends a right angle at the origin is

(A) x+y = 2 (B) x²+y² = 1 (C) x²+y² = 2 (D) x+y = 1

Q.15 Given the family of lines, a(2x + y + 4) + b(x − 2y − 3) = 0 . Among the lines of the family, the number of lines situated at a distance of 10 from the point M(2, −3) is :

(A) 0 (B) 1 (C) 2 (D) ∞

Q.16 The equation of the line passing through the points of intersection of the circles ; 3x²+3y²−2x+12y−9 = 0 & x²+y²+ 6x+ 2y−15 = 0 is :

(A) 10x−3y−18 = 0 (B) 5x+3y−18 = 0

(C) 5x−3y−18 = 0 (D) 10x+3y+1 = 0

Q.17 Through a point A on the x-axis a straight line is drawn parallel to y-axis so as to meet the pair of straight lines ax² + 2hxy + by² = 0 in B and C. If AB = BC then

(A) h² = 4ab (B) 8h² = 9ab (C) 9h² = 8ab (D) 4h² = ab Q.18 The number of common tangent(s) to the circles x² +y² +2x+8y –23 = 0 and

x² +y² – 4x –10y+19 = 0 is

(A) 1 (B) 2 (C) 3 (D) 4

Q.19 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then (A) ABC is a unique triangle (B) There can be only two such triangles.

(C) No such triangle is possible (D) There can be infinite number of such triangles.

Q.20 From the point A(0, 3) on the circle x² +4x+(y−3)² = 0 a chord AB is drawn & extended to a point M such that AM = 2AB. The equation of the locus of M is :

(A) x²+8x+ y² = 0 (B) x²+8x+ (y−3)² = 0 (C) (x−3)²+8x+y² = 0 (D) x²+8x+8y² = 0

Q.21 If A (1, p²) ; B (0, 1) and C (p, 0) are the coordinates of three points then the value of p for which the area of the triangle ABC is minimum, is

(A) 3 1

(B) – 3 1

(C) 3 1

or – 3 1

(D) none

Q.22 The area of the quadrilateral formed by the tangents from the point (4, 5) to the circle x²+ y²− 4x−2y−11 = 0 with the pair of radii through the points of contact of the tangents is : (A) 4 sq.units (B) 8 sq.units (C) 6 sq.units (D) none

Q.23 The area of triangle formed by the lines x + y – 3 = 0 , x – 3y + 9 = 0 and 3x – 2y + 1= 0 (A) 7

16 sq. units (B) 7

10 sq. units (C) 4 sq. units (D) 9 sq. units

Q.24 Two circles of radii 4cms & 1cm touch each other externally and θ is the angle contained by their direct common tangents. Then sinθ =

(A) 25

24 (B)

25

12 (C)

4

3 (D) none

Q.25 The set of lines ax + by + c = 0, where 3a + 2b + 4c = 0, is concurrent at the point : (A) ³

4 3 , 4



 

 (B) ¹

2 1 , 2



 

 (C) ³

4 1 , 2



 

 (D) (1, 1)

Quest

Q.26 The locus of poles whose polar with respect to x²+y² = a² always passes through (K, 0) is (A) Kx−a² = 0 (B) Kx+a² = 0 (C) Ky+a² = 0 (D) Ky−a² = 0

Q.27 The co−ordinates of the point of reflection of the origin (0, 0) in the line 4x − 2y − 5 = 0 is :

(A) (1, −2) (B) (2, −1) (C) ⁴

5 2 ,−5



 

 (D) (2, 5)

Q.28 The locus of the mid points of the chords of the circle x² + y² −ax−by = 0 which subtend a right angle at ^a

2 b , 2



 

 is

(A) ax+by = 0 (B) ax+by = a² +b²

(C) x² +y² −ax−by+ 8

b a²+ ²

= 0 (D) x² + y² −ax−by − 8

b a²+ ²

= 0

Q.29 A ray of light passing through the point A(1, 2) is reflected at a point B on the x−axis and then passes through (5, 3) . Then the equation of AB is :

(A) 5x + 4y = 13 (B) 5x − 4y = −3 (C) 4x + 5y = 14 (D) 4x − 5y = − 6

Q.30 From (3, 4) chords are drawn to the circle x² + y²−4x = 0. The locus of the mid points of the chords is (A) x² + y²−5x−4y + 6 = 0 (B) x² + y² + 5x−4y + 6 = 0

(C) x² + y²−5x + 4y + 6 = 0 (D) x² + y²−5x−4y−6 = 0

Q.31 m, n are integer with 0 < n < m. A is the point (m, n) on the cartesian plane. B is the reflection of A in the line y = x. C is the reflection of B in the y-axis, D is the reflection of C in the x-axis and E is the reflection of D in the y-axis. The area of the pentagon ABCDE is

(A) 2m(m + n) (B) m(m + 3n) (C) m(2m + 3n) (D) 2m(m + 3n)

Q.32 Which one of the following is false ?

The circles x²+ y²−6x−6y+ 9 = 0 & x²+y²+ 6x+ 6y+ 9 = 0 are such that : (A) they do not intersect

(B) they touch each other

(C) their exterior common tangents are parallel (D) their interior common tangents are perpendicular.

Q.33 The lines y − y₁ = m (x − x₁) ± a ₁+ m² are tangents to the same circle . The radius of the circle is

(A) a/2 (B) a (C) 2a (D) none

Q.34 The centre of the smallest circle touching the circles x² + y²−2y−3 = 0 and x² + y²−8x−18y + 93 = 0 is :

(A) (3, 2) (B) (4, 4) (C) (2, 7) (D) (2, 5)

Q.35 The ends of the base of an isosceles triangle are at (2, 0) and (0, 1) and the equation of one side is x = 2 then the orthocentre of the triangle is

(A) 



 



 2 ,3 4 3

(B) 



 



 ,1 4 5

(C) 



 



 ,1 4 3

(D) 



 



 12 , 7 3 4

Q.36 A rhombus is inscribed in the region common to the two circles x² + y² −4x−12 = 0 and x² + y² + 4x−12 = 0 with two of its vertices on the line joining the centres of the circles. The area of the rhombous is

(A) 8 3 sq.units (B) 4 3 sq.units (C) 16 3 sq.units (D) none

[5]

Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Q.37 A variable straight line passes through a fixed point (a, b) intersecting the co−ordinates axes at A & B. If 'O' is the origin then the locus of the centroid of the triangle OAB is :

(A) bx + ay − 3xy = 0 (B) bx + ay − 2xy = 0

(C) ax + by − 3xy = 0 (D) none

Q.38 The angle between the two tangents from the origin to the circle (x−7)² + (y+1)² = 25 equals (A) ^π

4 (B) ^π

3 (C) ^π

2 (D) none

Q.39 If P = (1, 0); Q = (−1, 0) & R = (2, 0) are three given points, then the locus of the points S satisfying the relation, SQ² + SR² = 2SP² is :

(A) a straight line parallel to x−axis (B) a circle passing through the origin (C) a circle with the centre at the origin (D) a straight line parallel to y−axis .

Q.40 The equation of the circle having normal at (3, 3) as the straight line y = x and passing through the point (2, 2) is :

(A) x² + y²−5x + 5y + 12 = 0 (B) x² + y² + 5x−5y + 12 = 0 (C) x² + y²−5x−5y−12 = 0 (D) x² + y²−5x−5y + 12 = 0

Q.41 The equation of the base of an equilateral triangle ABC is x + y = 2 and the vertex is (2, −1) . The area of the triangle ABC is :

(A) ²

6 (B) ³

6 (C) ³

8 (D) none

Q.42 In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord joining A with the point of intersection D of the hypotenuse and the semicircle, then the length AC equals to (A)

AB AD

AB AD

⋅ +

2 2 (B) AB AD

AB AD

⋅

+ ^(C) AB AD⋅ ^(D)

AB AD AB AD

⋅

−

2 2

Q.43 The equation of the pair of bisectors of the angles between two straight lines is,

12x² − 7xy − 12y² = 0. If the equation of one line is 2y − x = 0 then the equation of the other line is : (A) 41x − 38y = 0 (B) 38x − 41y = 0 (C) 38x + 41y = 0 (D) 41x + 38y = 0

Q.44 If the circle C₁ : x² + y² = 16 intersects another circle C₂ of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to 3/4, then the co-ordinates of the centre of C₂ are :

(A) ± ±

 

 9 5

12

, 5 (B) ±

 

 9 5

12

, ∓ 5 (C) ± ±

 

 12

5 9

, 5 (D) ±

 

 12

5 9 , ∓5 Q.45 Area of the rhombus bounded by the four lines, ax ± by ± c = 0 is :

(A) ^c ab

2

2 (B) ²

c2

ab (C) ⁴

c2

ab (D) ^ab

c 4 ²

Q.46 Two lines p₁x + q₁y + r₁ = 0 & p₂x + q₂y + r₂ = 0 are conjugate lines w.r.t. the circle x² + y² = a² if (A) p₁p₂ + q₁q₂ = r₁r₂ (B) p₁p₂ + q₁q₂ + r₁r₂ = 0

(C) a²(p₁p₂ + q₁q₂) = r₁r₂ (D) p₁p₂ + q₁q₂ = a²r₁r₂ Q.47 Area of the quadrilateral formed by the lines x + y = 2 is :

(A) 8 (B) 6 (C) 4 (D) none

Q.48 If the two circles (x−1)² + (y−3)² = r² & x²+y²−8x+2y + 8 = 0 intersect in two distinct points then (A) 2 < r < 8 (B) r < 2 (C) r = 2 (4) r > 2

Quest

Q.49 Let the algebraic sum of the perpendicular distances from the points (3, 0), (0, 3) & (2, 2) to a variable straight line be zero, then the line passes through a fixed point whose co-ordinates are :

(A) (3, 2) (B) (2, 3) (C) 3

5 3 , 5



 

 (D) 5

3 5 , 3



 



Q.50 If a circle passes through the point (a, b) & cuts the circle x²+ y² = K² orthogonally, then the equation of the locus of its centre is :

(A) 2ax+ 2by−(a²+ b²+ K²) = 0 (B) 2ax+ 2by−(a²−b²+K²) = 0

(C) x²+y²−3ax−4by+(a²+b²−K²) = 0 (D) x²+y²−2ax−3by+(a²−b²−K²) = 0 Q.51 Consider a quadratic equation in Z with parameters x and y as

Z² – xZ + (x – y)² = 0

The parameters x and y are the co-ordinates of a variable point P w.r.t. an orthonormal co-ordinate system in a plane. If the quadratic equation has equal roots then the locus of P is

(A) a circle

(B) a line pair through the origin of co-ordinates with slope 1/2 and 2/3 (C) a line pair through the origin of co-ordinates with slope 3/2 and 2 (D) a line pair through the origin of co-ordinates with slope 3/2 and 1/2

Q.52 Consider the circle S ≡ x² + y² – 4x – 4y + 4 = 0. If another circle of radius 'r' less than the radius of the circle S is drawn, touching the circle S, and the coordinate axes, then the value of 'r' is

(A) 3 – 2 2 ^{(B) 4 –} 2 2 ^{(C) 7 –} 4 2 ^{(D) 6 –} 4 2

Q.53 Vertices of a parallelogram ABCD are A(3, 1), B(13, 6), C(13, 21) and D(3, 16). If a line passing through the origin divides the parallelogram into two congruent parts then the slope of the line is (A) 12

11 (B)

8

11 (C)

8

25 (D)

8 13

Q.54 The distance between the chords of contact of tangents to the circle ; x²+y² +2gx+2fy+ c = 0 from the origin & the point (g, f) is :

(A) g²+f² (B) ^{g f c}

2 2

2

+ −

(C) ^{g f c} g f

2 2

2 2

2

+ −

+ (D) ^{g f c}

g f

2 2

2 2

2

+ +

+

Q.55 Two mutually perpendicular straight lines through the origin from an isosceles triangle with the line 2x + y = 5. Then the area of the triangle is

(A) 5 (B) 3 (C) 5/2 (D) 1

Q.56 The locus of the centers of the circles which cut the circles x² +y² +4x−6y + 9 = 0 and x² +y² −5x+4y−2 = 0 orthogonally is

(A) 9x+10y−7 = 0 (B) x−y+2 = 0 (C) 9x−10y+11=0 (D) 9x+10y+7 = 0 Q.57 Distance between the two lines represented by the line pair,

x² − 4xy + 4y² + x − 2y − 6 = 0 is : (A) ¹

5

(B) ₅ (C) 2 ₅ (D) none

Q.58 The locus of the center of the circles such that the point (2,3) is the mid point of the chord 5x+2y = 16 is :

(A) 2x−5y + 11 = 0 (B) 2x+5y−11 = 0

(C) 2x+5y+11 = 0 (D) none

[7]

Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Q.59 The distance between the two parallel lines is 1 unit. A point 'A' is chosen to lie between the lines at a distance 'd' from one of them. Triangle ABC is equilateral with B on one line and C on the other parallel line . The length of the side of the equilateral triangle is

(A) d d 1

3

2 ₂

+

+ (B)

3 1 d 2 d

2− +

(C) 2 d²−d+1 (D) d²−d+1

Q.60 The locus of the mid points of the chords of the circle x²+y²+4x−6y−12 = 0 which subtend an angle of ^π

3 radians at its circumference is :

(A) (x−2)² + (y + 3)² = 6.25 (B) (x + 2)² + (y−3)² = 6.25 (C) (x + 2)² + (y−3)² = 18.75 (D) (x + 2)² + (y + 3)² = 18.75

Q.61 Given A(0, 0) and B(x, y) with x ∈ (0, 1) and y > 0. Let the slope of the line AB equals m₁. Point C lies on the line x = 1 such that the slope of BC equals m₂ where 0 < m₂ < m₁. If the area of the triangle ABC can be expressed as (m₁ – m₂) f (x), then the largest possible value of f (x) is

(A) 1 (B) 1/2 (C) 1/4 (D) 1/8

Q.62 If two chords of the circle x² + y² − ax − by = 0, drawn from the point (a, b) is divided by the x−axis in the ratio 2 : 1 then:

(A) a² > 3b² (B) a² < 3b² (C) a² > 4 b² (D) a² < 4 b²

Q.63 P lies on the line y = x and Q lies on y = 2x. The equation for the locus of the mid point of PQ, if

| PQ | = 4, is

(A) 25x² + 36xy + 13y² = 4 (B) 25x² – 36xy + 13y² = 4 (C) 25x² – 36xy – 13y² = 4 (D) 25x² + 36xy – 13y² = 4 Q.64 The points (x₁, y₁), (x₂, y₂), (x₁, y₂) & (x₂, y₁) are always :

(A) collinear (B) concyclic

(C) vertices of a square (D) vertices of a rhombus

Q.65 If the vertices P and Q of a triangle PQR are given by (2, 5) and (4, –11) respectively, and the point R moves along the line N: 9x + 7y + 4 = 0, then the locus of the centroid of the triangle PQR is a straight line parallel to

(A) PQ (B) QR (C) RP (D) N

Q.66 The angle at which the circles (x – 1)² + y² = 10 and x² + (y – 2)² = 5 intersect is (A)

6

π (B)

4

π (C)

3

π (D)

2 π

Q.67 The co−ordinates of the points A, B, C are (−4, 0) , (0, 2) & (−3, 2) respectively. The point of intersection of the line which bisects the angle CAB internally and the line joining C to the middle point of AB is

(A) −

 

 7 3

4

, 3 (B) −

 

 5 2

13

, 2 (C) ⁷

3 10 ,− 3



 

 (D) −

 

 5 2

3 ,2

Q.68 Two congruent circles with centres at (2, 3) and (5, 6) which intersect at right angles has radius equal to

(A) 2 2 ^{(B) 3 (C) 4 (D) none}

Q.69 Three lines x + 2y + 3 = 0 ; x + 2y – 7 = 0 and 2x – y – 4 = 0 form the three sides of two squares. The equation to the fourth side of each square is

(A) 2x – y + 14 = 0 & 2x – y + 6 = 0 (B) 2x – y + 14 = 0 & 2x – y – 6 = 0 (C) 2x – y – 14 = 0 & 2x – y – 6 = 0 (D) 2x – y – 14 = 0 & 2x – y + 6 = 0

Quest

Q.70 A circle of radius unity is centred at origin. Two particles start moving at the same time from the point (1, 0) and move around the circle in opposite direction. One of the particle moves counterclockwise with constant speed v and the other moves clockwise with constant speed 3v. After leaving (1, 0), the two particles meet first at a point P, and continue until they meet next at point Q. The coordinates of the point Q are

(A) (1, 0) (B) (0, 1) (C) (0, –1) (D) (–1, 0)

Q.71 The points A(a, 0), B(0, b), C(c, 0) & D(0, d) are such that ac = bd & a, b, c, d are all non−zero. The the points

(A) form a parallelogram (B) do not lie on a circle

(C) form a trapezium (D) are concyclic

Q.72 The value of 'c' for which the set, {(x, y)x² + y² + 2x ≤ 1} ∩ {(x, y)x − y + c ≥ 0} contains only one point in common is :

(A) (−∞, −1] ∪ [3, ∞) (B) {−1, 3}

(C) {−3} (D) {−1 }

Q.73 Given A ≡ (1, 1) and AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and meets the y-axis in C, then the equation of locus of mid- point P of BC is

(A) x + y = 1 (B) x + y = 2 (C) x + y = 2xy (D) 2x + 2y = 1

Q.74 A circle is inscribed into a rhombous ABCD with one angle 60º. The distance from the centre of the circle to the nearest vertex is equal to 1 . If P is any point of the circle, then

PA²+ PB² + PC² + PD² is equal to :

(A) 12 (B) 11 (C) 9 (D) none

Q.75 The number of possible straight lines , passing through (2, 3) and forming a triangle with coordinate axes, whose area is 12 sq. units , is

(A) one (B) two (C) three (D) four

Q.76 P is a point (a, b) in the first quadrant. If the two circles which pass through P and touch both the co-ordinate axes cut at right angles, then :

(A) a² − 6ab + b² = 0 (B) a² + 2ab − b² = 0 (C) a² − 4ab + b² = 0 (D) a² − 8ab + b² = 0

Q.77 In a triangle ABC , if A (2, – 1) and 7x – 10y + 1 = 0 and 3x – 2y + 5 = 0 are equations of an altitude and an angle bisector respectively drawn from B, then equation of BC is

(A) x + y + 1 = 0 (B) 5x + y + 17 = 0 (C) 4x + 9y + 30 = 0 (D) x – 5y – 7 = 0

Q.78 The range of values of 'a' such that the angle θ between the pair of tangents drawn from the point (a, 0) to the circle x² + y² = 1 satisfies ^π

2 < θ < π is :

(A) (1, 2) (B)

( )

^{1, 2 (C)}

(

^{− 2 ,−1}

)

^(D)

(

^{− 2,−1}

)

^∪

( )

^{1, 2}

Q.79 Distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to the line 3x − 4y + 8 = 0 is

(A) 15/2 (B) 9/2 (C) 5 (D) None

Q.80 Three concentric circles of which the biggest is x² + y² = 1, have their radii in A.P. If the line y = x + 1 cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is (A) 0 1

, 4



 

 (B) 0 1

, 2 2



 

 (C) 0 2 2

, −4



 

 (D) none

[9]

Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Q.81 The co-ordinates of the vertices P, Q, R & S of square PQRS inscribed in the triangle ABC with vertices A ≡ (0, 0) , B ≡ (3, 0) & C ≡ (2, 1) given that two of its vertices P, Q are on the side AB are respectively Q.83 AB is the diameter of a semicircle k, C is an arbitrary point on the semicircle (other than A or B) and S is the centre of the circle inscribed into triangle ABC, then measure of

(A) angle ASB changes as C moves on k.

(B) angle ASB is the same for all positions of C but it cannot be determined without knowing the radius.

(C) angle ASB = 135° for all C.

(D) angle ASB = 150° for all C.

Q.84 Tangents are drawn to the circle x² + y² = 1 at the points where it is met by the circles,

x² + y² − (λ + 6)x + (8 − 2λ) y − 3 = 0 . λ being the variable . The locus of the point of intersection of these tangents is :

(A) 2x−y+10 = 0 (B) x+2y−10 = 0 (C) x−2y+10 = 0 (D) 2x+y−10 = 0 Q.85 Given ^x

a y

+b = 1 and ax + by = 1 are two variable lines, 'a' and 'b' being the parameters connected by the relation a² + b² = ab. The locus of the point of intersection has the equation

(A) x² + y² + xy − 1 = 0 (B) x² + y² – xy + 1 = 0 (C) x² + y² + xy + 1 = 0 (D) x² + y² – xy – 1 = 0

Q.86 B & C are fixed points having co−ordinates (3, 0) and (−3, 0) respectively. If the vertical angle BAC is 90º, then the locus of the centroid of the ∆ ABC has the equation :

(A) x² + y² = 1 (B) x² + y² = 2 (C) 9(x² + y²) = 1 (D) 9(x² + y²) = 4

Q.87 The set of values of 'b' for which the origin and the point (1, 1) lie on the same side of the straight line, a²x + aby + 1 = 0 ∀ a ∈ R, b > 0 are :

 are four distinct points on a circle of radius 4 units then, abcd is equal to

(A) 4 (B) 1/4 (C) 1 (D) 16

Q.89 Triangle formed by the lines x + y = 0 , x – y = 0 and lx + my = 1. If l and m vary subject to the condition l ² + m² = 1 then the locus of its circumcentre is

(A) (x² – y²)² = x² + y² (B) (x² + y²)² = (x² – y²) (C) (x² + y²) = 4x² y² (D) (x² – y²)² = (x² + y²)²

Q.90 Tangents are drawn to a unit circle with centre at the origin from each point on the line 2x + y = 4. Then the equation to the locus of the middle point of the chord of contact is

(A) 2 (x² + y²) = x + y (B) 2 (x² + y²) = x + 2y (C) 4 (x² + y²) = 2x + y (D) none

Quest

Q.91 The co−ordinates of three points A(−4, 0) ; B(2, 1) and C(3, 1) determine the vertices of an equilateral trapezium ABCD. The co−ordinates of the vertex D are :

(A) (6, 0) (B) (−3, 0) (C) (−5, 0) (D) (9, 0)

Q.92 ABCD is a square of unit area. A circle is tangent to two sides of ABCD and passes through exactly one of its vertices. The radius of the circle is

(A) 2− 2 ^(B) 2−1 ^{(C) 1}₂ ^(D) 2

1

Q.93 A parallelogram has 3 of its vertices as (1, 2), (3, 8) and (4, 1). The sum of all possible x-coordinates for the 4^th vertex is

(A) 11 (B) 8 (C) 7 (D) 6

Q.94 A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at A enclosing an angle of 60°. The area enclosed by these tangents and the arc of the circle is

(A) 3 2

– 6

π (B) 3 –

3

π (C)

3 π –

6

3 (D) 



 



 −π 1 6 3

Q.95 The image of the pair of lines represented by ax² + 2h xy + by² = 0 by the line mirror y = 0 is (A) ax² − 2h xy − by² = 0 (B) bx² − 2h xy + ay² = 0

(C) bx² + 2h xy + ay² = 0 (D) ax² − 2h xy + by² = 0

Q.96 A straight line with slope 2 and y-intercept 5 touches the circle, x² + y² + 16x + 12y + c = 0 at a point Q. Then the coordinates of Q are

(A) (–6, 11) (B) (–9, –13) (C) (–10, – 15) (D) (–6, –7)

Q.97 The acute angle between two straight lines passing through the point M(−6, −8) and the points in which the line segment 2x + y + 10 = 0 enclosed between the co-ordinate axes is divided in the ratio 1 : 2 : 2 in the direction from the point of its intersection with the x-axis to the point of intersection with the y-axis is

(A) π/3 (B) π/4 (C) π/6 (D) π/12

Q.98 A variable circle cuts each of the circles x² + y² − 2x = 0 & x² + y² − 4x − 5 = 0 orthogonally. The variable circle passes through two fixed points whose co−ordinates are :

(A) ^{− ±}

 



5 3

2 ,0 (B) ^{− ±}



 



5 3 5

2 ,0 (C) ^{− ±}



 



5 5 3

2 ,0 (D) ^{− ±}



 



5 5

2 ,0 Q.99 If in triangle ABC, A ≡ (1, 10) , circumcentre ≡

(

^{− 1}3 2

)

, 3 and orthocentre ≡

(

¹¹3 4

)

, 3 then the co-ordinates of mid-point of side opposite to A is :

(A) (1, −11/3) (B) (1, 5) (C) (1, −3) (D) (1, 6)

Q.100 The radical centre of three circles taken in pairs described on the sides of a triangle ABC as diametres is the (A) centroid of the ∆ ABC (B) incentre of the ∆ ABC

(C) circumcentre o the ∆ ABC (D) orthocentre of the ∆ ABC

Q.101 The line x + y = p meets the axis of x & y at A & B respectively . A triangle APQ is inscribed in the triangle OAB, O being the origin, with right angle at Q . P and Q lie respectively on OB and AB. If the area of the triangle APQ is 3/8^th of the area of the triangle OAB, then ^{A Q}

BQ is equal to :

(A) 2 (B) 2/3 (C) 1/3 (D) 3

[11]

Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Q.102 Two circles are drawn through the points (1, 0) and (2, −1) to touch the axis of y. They intersect at an angle (A) cot^{–1 3}

4 (B) cos ^{−1 4}

5 (C) ^π

2 (D) tan⁻¹ 1

Q.103 In a triangle ABC, side AB has the equation 2x + 3y = 29 and the side AC has the equation, x + 2y = 16 . If the mid−point of BC is (5, 6) then the equation of BC is :

(A) x − y = − 1 (B) 5x − 2y = 13 (C) x + y = 11 (D) 3x − 4y = − 9

Q.104 If the line x cosθ + y sinθ = 2 is the equation of a transverse common tangent to the circles x² + y² = 4 and x² + y² − 6 3 x − 6y + 20 = 0, then the value of θ is :

(A) 5π/6 (B) 2π/3 (C) π/3 (D) π/6

Q.105 ABC is an isosceles triangle. If the co-ordinates of the base are (1, 3) and (− 2, 7), then co-ordinates of vertex A can be :

(A)

(

⁻¹2 , 5

)

(B)

(

⁻¹8 , 5

)

(C)

(

⁵₆ ^{, −5}

)

^(D)

(

^{− 7}, ¹8

)

Q.106 A circle is drawn with y-axis as a tangent and its centre at the point which is the reflection of (3, 4) in the line y = x. The equation of the circle is

(A) x² + y² – 6x – 8y + 16 = 0 (B) x² + y² – 8x – 6y + 16 = 0 (C) x² + y² – 8x – 6y + 9 = 0 (D) x² + y² – 6x – 8y + 9 = 0 Q.107 A is a point on either of two lines y + ₃x = 2 at a distance of ⁴

3 units from their point of intersection.

The co-ordinates of the foot of perpendicular from A on the bisector of the angle between them are (A) −

 

 2

3 ,2 (B) (0, 0) (C) ²

3 ,2



 

 (D) (0, 4)

Q.108 A circle of constant radius 'a' passes through origin 'O' and cuts the axes of co−ordinates in points P and Q, then the equation of the locus of the foot of perpendicular from O to PQ is :

(A) (x² + y²) ¹₂ ¹₂ x + y



 

 _{= 4a}² _{(B) (x}² _{+ y}²₎^{2 1 1}

2 2

x + y



 

 _{= a}²

(C) (x² + y²)^{2 1}₂ ¹₂ x + y



 

 _{= 4a}² _{(D) (x}² _{+ y}²₎ ^{1 1}

2 2

x + y



 

 _{= a}²

Q.109 Three straight lines are drawn through a point P lying in the interior of the ∆ ABC and parallel to its sides.

The areas of the three resulting triangles with P as the vertex are s₁, s₂ and s₃. The area of the triangle in terms of s₁, s₂ and s₃ is :

(A) s s_{1 2} +s s_{2 3} +s s_{3 1} (B) ³s s s_{1 2 3}

(C)

(

^{s1 + s2 + s3}

)

^{2 (D) none}

Q.110 The circle passing through the distinct points (1, t), (t, 1) & (t, t) for all values of 't' , passes through the point :

(A) (−1, −1) (B) (−1, 1) (C) (1, −1) (D) (1, 1)

Q.111 The sides of a ∆ ABC are 2x − y + 5 = 0 ; x + y − 5 = 0 and x − 2y − 5 = 0 . Sum of the tangents of its interior angles is :

(A) 6 (B) 27/4 (C) 9 (D) none

Quest

Q.112 If a circle of constant radius 3k passes through the origin 'O' and meets co-ordinate axes at A and B then the locus of the centroid of the triangle OAB is

(A) x² + y² = (2k)² (B) x² + y² = (3k)² (C) x² + y² = (4k)² (D) x² + y² = (6k)²

Q.113 Chords of the curve 4x² + y² − x + 4y = 0 which subtend a right angle at the origin pass through a fixed point whose co-ordinates are

(A) ¹ 5

4 ,−5



 

 (B) ^_⁻ 

 1 5

4

, 5 (C) ¹

5 4 , 5



 

 (D) ^_^{− −} 

 1 5

4 , 5

Q.114 Let x & y be the real numbers satisfying the equation x² − 4x + y² + 3 = 0. If the maximum and minimum values of x² + y² are M & m respectively, then the numerical value of M − m is :

(A) 2 (B) 8 (C) 15 (D) none of these

Q.115 If the straight lines joining the origin and the points of intersection of the curve 5x² + 12xy − 6y² + 4x − 2y + 3 = 0 and x + ky − 1 = 0

are equally inclined to the co-ordinate axes then the value of k :

(A) is equal to 1 (B) is equal to −1

(C) is equal to 2 (D) does not exist in the set of real numbers .

Q.116 A line meets the co-ordinate axes in A & B. A circle is circumscribed about the triangle OAB. If d₁ & d₂ are the distances of the tangent to the circle at the origin O from the points A and B respectively, the diameter of the circle is :

(A) 2 d d 2 ₁+ ₂

(B) 2 d 2 d₁+ ₂

(C) d₁ + d₂ (D)

2 1

2 1

d d

d d

+

Q.117 A pair of perpendicular straight lines is drawn through the origin forming with the line 2x + 3y = 6 an

Q.117 A pair of perpendicular straight lines is drawn through the origin forming with the line 2x + 3y = 6 an

In document Maths+quest (Page 103-135)