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Q.1. Let the tangents drawn from the origin to the circle, x^{2}+ y^{2}  8x  4y + 16 = 0 touch it at the points A and B. The (AB)^{2} is equal to (2020)
(1) 52/5
(2) 56/5
(3) 64/5
(4) 32/5
Ans. (3)
Solution. The equation of the given circle is
x^{2}+ y^{2}  8x  4y + 16 = 0
Therefore,
Radius of the circle, R =
Length of tangent, L = √S_{1} = √16 = 4
Now, the length of chord AB =
Hence, the square of length of chord is
Q.2. If a line y = mx + c is a tangent to the circle (x  3)^{2} + y^{2} = 1 and it is perpendicular to a line L_{1}, where L_{1} is the tangent to the circle x^{2} + y^{2} = 1 at the point then (2020)
(1) c^{2}  7c + 6 = 0
(2) c^{2} + 7c + 6 = 0
(3) c^{2} + 6c + 7 = 0
(4) c^{2}  6c + 7 = 0
Ans. (3)
Solution.
Given, L_{1} is the tangent to the circle x^{2} + y^{2} = 1 at point (1/√2, 1/√2); so
Now, the slope of line y = mx + c is
m = 1/m_{L} = 1
Therefore, y = x + c is tangent to (x  3)^{2} + y^{2} = 1, so the length of perpendicular from the center (3, 0) to the tangent is equal to the radius of the circle.
Now,
⇒ c^{2} + 6c + 7 = 0
Q.3. A circle touches the yaxis at the point (0, 4) and passes through the point (2, 0). Which of the following lines is not a tangent to this circle? (2020)
(1) 4x + 3y + 17 =0
(2) 3x + 4y  24 = 0
(3) 3x + 4y  6 = 0
(4) 4x + 3y  8 = 0
Ans. (4)
Solution. We have
Therefore, the equation of the circle is
(x  0)^{2} + (y  4)^{2} + λx = 0 ...(1)
It passes through the (2,0) , then
4 + 16 + 2λ = 0 ⇒ λ =  10
The equation of the circle is
x^{2} + y^{2}  10x + 8y + 16 = 0
Hence, the radius of the circle is
⇒
Now, the lengths of perpendicular drawn from the center (5,4) to the given lines are
Q.4. If the curves x^{2}  6x + y^{2} + 8 = 0 and x^{2}  8y + y^{2} +16  k = 0, (k > 0) touch each other at a point, then the largest value of k is ________. (2020)
Ans. (36.00)
Solution. For the circle x^{2}  6x + y^{2} +8 = 0, we have
C_{1} = (3, 0) and r_{1} = = 1
For the circle x^{2}  8y + y^{2} + 16  k = 0, we have
Two circles touch each other if C_{1}C_{2} = r_{1} ± r_{2}
⇒ k = 16 or 36
Q.5. Equation of a common tangent to the circle, x^{2} + y^{2}  6x = 0 and the parabola, y^{2} = 4x, is: (2019)
(1)
(2)
(3)
(4)
Ans. (2)
Solution. Since, the equation of tangent to parabola y^{2} = 4x is
...(1)
The line (1) is also the tangent to circle
x^{2 }+ y^{2 } 6x = 0
Then centre of circle = (3, 0)
Radius of circle = 3
The perpendicular distance from centre to tangent is equal to the radius of circle
⇒
Hence, = x + 3 is one of the required common tangent.
Q.6. Three circles of radii a, b, c (a < b < c) touch each other externally. If they have xaxis as a common tangent, then: (2019)
(1)
(2)
(3) a, b, c are in A.P
(4)
Ans. (1)
Solution.
Q.7. If the circles x^{2} + y^{2}  16x  20y + 164 = r^{2} and (x  4) + (y  7)^{2} = 36 intersect at two distinct points, then: (2019)
(1) r > 11
(2) 0 < r < 1
(3) r = 11
(4) 1 < r < 11
Ans. (4)
Solution. Consider the equation of circles as,
Both the circles intersect each other at two distinct points. Distance between centres
Q.8. If a circle C passing through the point (4, 0) touches the circle x^{2} + y^{2} + 4x  6y = 12 externally at the point (1, 1), then the radius of C is: (2019)
(1) 2√5
(2) 4
(3) 5
(4) √57
Ans. (3)
Solution. The equation of circle x^{2} + y^{2} + 4x  6y = 12 can be written as (x + 2)^{2} + (y  3)^{2} = 25
Let P = (1, 1) & Q = (4, 0)
Equation of tangent at P (1, 1) to the given circle:
x(1) + y(1) + 2(x+1)  3(y1)  12 = 0
3x  4y  7 = 0 ...(1)
The required circle is tangent to (1) at (1, 1).
∴ (x1)^{2} + (y+1)2 + λ(3x4y7) = 0 ...(2)
Equation (2) passes through Q(4, 0)
⇒ 3^{2}+1^{2}+λ(127) = 0 ⇒ 5λ+10 = 0
⇒ λ = 2
Equation (2) becomes x^{2} + y^{2}  8x + 10y + 16 = 0
Q.9. If the area of an equilateral triangle inscribed in the circle, x^{2} + y^{2} + 10x + 12y + c = 0 is sq. units then c is equal to: (2019)
(1) 13
(2) 20
(3) 25
(4) 25
Ans. (4)
Solution.
Let the sides of equilateral Δ inscribed in the circle be a, then
Then, area of the equilateral triangle
But it is given that area of equilateral triangle
 constant term = r^{2}
(5)^{2} + (6)^{2}  c = 36
c = 25
Q.10. A square is inscribed in the circle x^{2} + y^{2}  6x + 8y  103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is: (2019)
(1) 6
(2)
(3)
(4) 13
Ans. (3)
Solution. The equation of circle is,
Q.11. Two circles with equal radii are intersecting at the points (0, 1) and (0, 1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is: (2019)
(1) 1
(2) 2
(3) 2√2
(4) √2
Ans. (2)
Solution.
∵ Two circles of equal radii intersect each other orthogonally. Then R is mid point of PQ.
∴ Distance between centres = 1 + 1 = 2.
Q.12. A circle cuts a chord of length 4a on the xaxis and passes through a point on the yaxis, distant 2b from the origin. Then the locus of the centre of this circle, is: (2019)
(1) a hyperbola
(2) an ellipse
(3) a straight line
(4) a parabola
Ans. (4)
Solution.
Hence, the above locus of the centre of circle is a parabola.
Q.13. Let C_{1} and C_{2} be the centres of the circles x^{2} + y^{2}  2x  2y  2 = 0 and x^{2} + y^{2 } 6x  6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles then, the area (in sq. units) of the quadrilateral PC_{1}QC_{2} is: (2019)
(1) 8
(2) 6
(3) 9
(4) 4
Ans. (4)
Solution.
Hence, circles intersect orthogonally
∴ Area of the quadrilateral PC_{1}QC_{1}
Q.14. If a variable line, 3x + 4y  λ = 0 is such that the two circles x^{2} + y^{2 } 2x  2y + 1 =0 and x^{2} + y^{2} 18x  2y + 78 = 0 are on its opposite sides, then the set of all values of λ is the interval: (2019)
(1) (2,17)
(2) [13,23]
(3) [12,21]
(4) (23,31)
Ans. (3)
Solution.
Condition 1: The centre of the two circles are (1, 1) and (9, 1). The circles are on opposite sides of the line 3x + 4y  λ = 0.
Put x = 1, y = 1 in the equation of line,
Now, put x = 9, y = 1 in the equation of line,
Condition 2: Perpendicular distance from centre on line ≥ radius of circle.
Intersection of (1), (2) and (3) gives λ ∈ [12, 21].
Q.15. If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is: (2019)
(1)
(2)
(3)
(4)
Ans. (2)
Solution. As ∠AOB = 90°
Let AB diameter and M(h, k) be foot of perpendicular, then
Then, equation of AB
∴ AB is the diameter, then
AB = 2R
⇒ AB^{2} = 4R^{2}
Hence, required locus is (x^{2} + y^{2})^{3} = 4R^{2} x^{2} y^{2 }
Q.16. The sum of the squares of the lengths of the chords intercepted on the circle, x^{2} + y^{2} = 16, by the lines, x + y = n, n∈N, where N is the set of all natural numbers, is: (2019)
(1) 320
(2) 105
(3) 160
(4) 210
Ans. (4)
Solution. Let the chord x + y = n cuts the circle x^{2} + y^{2} = 16 at P and Q Length of perpendicular from O on PQ
Then, length of chord
Thus only possible values of n are 1, 2, 3, 4, 5.
Hence, the sum of squares of lengths of chords
Q.17. The tangent and the normal lines at the point (√3, 1) to the circle x^{2} + y^{2} = 4 and the xaxis form a triangle. The area of this triangle (in square units) is: (2019)
(1)
(2) 1/3
(3)
(4)
Ans. (3)
Solution.
Q.18. If a tangent to the circle x^{2} + y^{2}  1 intersects the coordinate axes at distinct points P and Q, then the locus of the midpoint of PQ is: (2019)
(1) x^{2} + y^{2}  4x^{2}y^{2} = 0
(2) x^{2} + y^{2}  2xy = 0
(3) x^{2} + y^{2}  16x^{2}y^{2} = 0
(4) x^{2} + y^{2}  2x^{2}y^{2} = 0
Ans. (1)
Solution. Let any tangent to circle x^{2} + y^{2} = 1 is
x cosθ + y sinθ = 1
Since, P and Q are the point of intersection on the coordinate axes.
Now squaring and adding equation (1) and (2)
Q.19. The common tangent to the circles x^{2} + y^{2} = 4 and x^{2} + y^{2} + 6x + 8y  24 = 0 also passes through the point: (2019)
(1) (4,2)
(2) (6,4)
(3) (6,2)
(4) (4,6)
Ans. (3)
Solution. By the diagram,
Equation of common tangent is,
S_{1}  S_{2} = 0
6x + 8y  20 = 0
⇒ 3x + 4y  10 = 0
Hence (6, 2) lies on it.
Q.20. If the circles x^{2} + y^{2} + 5Kx + 2y + K = 0 and 2(x^{2 }+ y^{2}) + 2Kx + 3y  1=0, (K∈R), intersect at the points P and Q, then the line 4x + 5y  K = 0 passes through P and Q, for: (2019)
(1) infinitely many values of K
(2) no value of K.
(3) exactly two values of K
(4) exactly one value of K
Ans. (2)
Solution.
...(1)
Equation of the line passing through the intersection points P & O is.
4x + 5y  K = 0 ...(2)
Comparing (1) and (2),
...(3)
∴ No value of K exists.
Q.21. The line x = y touches a circle at the point (1, 1). If the circle also passes through the point (1, 3), then its radius is: (2019)
(1) 3
(2)
(3) 2
(4)
Ans. (2)
Solution. Equation of circle which touches the line y = x at (1,1) is, (x1)^{2} + (y1)^{2} + λ(yx) = 0
This circle passes through (1,3)
Hence, equation of circle will be,
Q.22. The locus of the centres of the circles, which touch the circle, x^{2} + y^{2} = 1 externally, also touch the yaxis and lie in the first quadrant, is: (2019)
(1)
(2)
(3)
(4)
Ans. (2)
Solution. Let centre of required circle is (h, k).
∴ OO' = r + r' [By the diagram]
Q.23. If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is: (2019)
(1) 13/5
(2) 120/13
(3) 60/13
(4) 13/2
Ans. (2)
Solution.
According to the diagram,
Q.24. A circle touching the xaxis at (3,0) and making an intercept of length 8 on the yaxis passes through the point: (2019)
(1) (3,10)
(2) (3,5)
(3) (2,3)
(4) (1,5)
Ans. (1)
Solution. Let centre of circle is C and circle cuts the vaxis at B and A. Let midpoint of chord BA is M.
equation of circle is,
(x  3)^{2} + (y  5)^{2} = 5^{2}
(3, 10) satisfies this equation.
Although there will be another circle satisfying the same conditions that will lie below the xaxis having equation (x  3)^{2} + (y  5)^{2} = 5^{2 }
Q.25. Let the orthocenter and centroid of a triangle be A(3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, than the radius of the circle having line segment AC as diameter, is: (2018)
(1) √10
(2) 2√10
(3)
(4)
Ans. (3)
Solution. Orthocentre A (3, 5) centroid B (3, 3)
Q.26. If the tangent at (1,7) to the curve x^{2} = y  6 touches the circle x^{2} + y^{2} + 16x + 12y + c = 0 than the value of c is: (2018)
(1) 195
(2) 185
(3) 85
(4) 95
Ans. (4)
Solution. Equation tangent at (1, 7)
⇒ 2x  y + 5 = 0
perpendicular (8, 6) to line
Q.27. A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, y  4x + 3 = 0, then its radius is equal to: (2018)
(1) √5
(2) 2
(3) √2
(4) 1
Ans. (2)
Solution. Let centre of circle is c(α,β)
it lies is line y – 4x + 3 = 0 B = 4α – 3
∴ c(α, 4α–3)
Q.28. If a circle C, whose radius is 3, touches externally the circle, x^{2} + y^{2} + 2x – 4y – 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle C, on the xaxis is equal to: (2018)
(1) 2√3
(2) √5
(3) 3√2
(4) 2√5
Ans. (4)
Solution.
Q.29. If a point P has co  ordinates ( 0,–2) and Q ia any point on the circle, x^{2 }+ y^{2 }– 5x – y + 5 = 0, then the maximum value of (PQ)^{2} is: (2017)
Ans. (3)
Solution.
The center of the circle and radius is
Let us put the values of the corrdinates of P(0,2) in the LHS of the equation of circle , we get and as it is is greater than 0, the point is outside the circle and maximum distance of P from any point on the circle would be, distance of P from centrer of circle plus radius.
Q.30. If two parallel chords of the a circle, having diameter 4 units, lie on the opposite sides of the centre and subtend angles and sec^{–1 }(7) at the centre respectively, then the distance between these chords, is: (2017)
(2) 16/7
(4) 8/7
Ans. (1)
Solution.
Distance between chords
= Radius Cos(subtended angle/2) by chord1 + Radius Cos(subtended angle/2) by chord 2
Radius = Diameter /2 = 4/2 = 2
= 2 Cos (θ₁/2) + 2Cos(θ₂/2)
θ₁ = Cos⁻¹(1/7) => Cosθ₁ = 1/7
θ₂ = Sec⁻¹7 => Secθ₂ = 7 => 1/Cosθ₂ = 7 => Cosθ₂ = 1/7
Applying Cos2θ = 2Cos²θ  1 => Cos²θ = (1 + cos2θ)/2
Putting θ = θ₁/2
Cos²(θ₁/2) = ( 1 + 1/7)/2 = 4/4
=> Cos(θ₁/2) = 2/√7
Putting θ = θ₂/2
Cos²(θ₂/2) = ( 1 + 1/7)/2 = 4/7
=> Cos(θ₁/2) = 2/√7
2 Cos (θ₁/2) + 2Cos(θ₂/2) = 2 *2/√7 + 2 * 2/√7
= 4/√7 + 4/√7
= 8/√7
Q.31. A line drawn through the point P(4,7) cuts the circle x^{2}+y^{2} = 9 at the points A and B. Then PA.PB is equal to: (2017)
(1) 74
(2) 53
(3) 56
(4) 65
Ans. (3)
Solution.
Length of a tangent from external point (x_{1},y_{1}) =
Q.32. The centres of those circles which touch the circle, x^{2} + y^{2}  8x  8y  4 = 0, externally and also touch the xaxis, lie on: (2016)
(1) a circle
(2) an ellipse which is not a circle
(3) a hyperbola
(4) a parabola
Ans. (4)
x^{2} + y^{2}  8x  8y  4 = 0
Centre (4, 4)
Radius = = 6
Let centre of the circle is (h, k)
= (6 + k)
(h  4)^{2} + (k  4)^{2} = (6 + k)^{2}
h^{2}  8h + 16 + k^{2}  8k + 16 = 36 + k^{2} + 12k
h^{2}  8h  20k  4 = 0
x^{2}  8x  20y  4 = 0
Which is an equation of parabola
Q.33. If one of the diameters of the circle, given by the equation, x^{2} + y^{2}  4x + 6y  12 = 0, is a chord of a circle S, whose centre is at (3, 2), then the radius of S is: (2016)
(1) 5√2
(2) 5√3
(3) 5
(4) 10
Ans. (2)
x^{2} + y^{2}  4x + 6y  12 = 0
Centre (2, 3)
Radius = 5
Distance b/w two centres c_{1}(2, 3) and c_{2}(3, 2)
d = = √50
Radius of (S) =
Q.34. A circle passes through (2, 4). Which one of the following equations can represent a diameter of this circle? (2016)
(1) 4x + 5y  6 = 0
(2) 5x + 2y + 4 = 0
(3) 2x  3y + 10 = 0
(4) 3x + 4y  3 = 0
Ans. (3)
Required circle is
(x  0)^{2} + (y  2)^{2} + λ(x) = 0
it passes (2, 4)
∴ 4 + 4  2λ = 0
λ = 4
∴ circle is x^{2} + y^{2}  4y + 4x + 4 = 0
centre (2, 2) which satisfy
2x  3y + 10 = 0
Q.35. Equation of the tangent to the circle, at the point (1, 1), whose centre is the point of intersection of the straight lines x  y = 1 and 2x + y = 3 is (2016)
(1) 3x  y  4 = 0
(2) x + 4y + 3 = 0
(3) x  3y  4 = 0
(4) 4x + y  3 = 0
Ans. (2)
Centre of circle is
⇒ equation of circle is
Equation of tangent at (1, 1) is 3x  3y  4(x + 1)  (y  1) = 0
⇒  x  4y  3 = 0
⇒ x + 4y + 3 = 0
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