Question 1

Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) The circle with centre (1, 2) and touching the straight line 3x + 4y = 1, passes through (1

Accepted Answer

For (P): The radius of the circle is the perpendicular distance from (1, 2) to 3x + 4y - 1 = 0. r = |3(1) + 4(2) - 1| 3^2 + 4^2 = 10 5 = 2 The equation of the circle is (x - 1)^2 + (y - 2)^2 = 4. Checking the given points, (3, 2) satisfies the equation. Thus, (P) (3). For (Q): The equation of a tangent to y^2 = 8x with slope m is y = mx + 2 m . Since it is also a tangent to x^2 + y^2 = 2, the perpendicular distance from (0, 0) to the line mx - y + 2 m = 0 is equal to the radius 2 . | 2 m | m^2 +

Question 2

Let the straight line y =2 x touch a circle with center (0, ), >0, and radius r at a point A _1. Let B _1 be the point on the circle such that the line segment A_1 B_1 is a diameter of the circle. Let

Accepted Answer

Consider centre as P (0, ), >0 | 2(0)- 5 |=r aligned & |- |= 5 r \& = 5 r \& +r=5+ 5 \& 5 r+r= 5 ( 5 +1) \& r= 5 , =5 \& P(0,5) aligned Foot of perpendicular from P to line 2 x - y =0 aligned & x-0 2 = y-5 -1 = -(2(0)-5) 5 =1 \& x=2, y=4 A_1(2,4) aligned Let B ( p , q ) p +2 2 =0, q +4 2 =5 p =-2, q =6 ~B (-2,6)

Question 3

Let C 1  be the circle of radius 1 with center at the origin. Let C 2 be the circle of radius r with center at the point A = ( 4 , 1 ) , where 1 &#60; r &#60; 3 . Two distinct common tangents P Q

Accepted Answer

Plotting the diagram of the given value we get, Now let M  and N be midpoints of P Q and S T respectively. &#8658; M N is a radical axis of two circles C 1 : x 2 + y 2 = 1 C 2 : ( x - 4 ) 2 + ( y - 1 ) 2 = r 2 &#8658; x 2 + y 2 - 8 x - 2 y + 17 - r 2 = 0 Now subtracting equation of both above circles we get, Equation of M N : &#160; 8 x + 2 y - 18 + r 2 = 0 Now the line  M N  intersect the  x - axis  at  B So,  B  will be, &#8658; B 18 - r 2 8 , 0 Als

Question 4

Let G be a circle of radius R &#62; 0 . Let G 1 , G 2 , &#8230; , G n be n circles of equal radius r &#62; 0 . Suppose each of the n circles G 1 , G 2 , &#8230; , G n touches the circle G externally.

Accepted Answer

Plotting the diagram of given condition we have,   Now each radius of smaller circle will make  &#960; n  angle at centre of bigger circle, Now applying the sine rule, we get sin &#960; n sin &#960; 2 = r R + r &#8658; R r + 1 = cosec &#960; n &#8658; R = r cosec &#960; n - 1 Now checking all options we get, (A) n = 4 , R = r 2 - 1 (B) n = 5 , R = r cosec &#960; 5 - 1 &#8658; R &#60; r cosec &#960; 6 - 1 &#8658; R &#60; r (C) n = 8 , R = r cosec &#960; 8 - 1 &#8658; R &#62; r cose

Question 5

Let the circles C 1 : x 2 + y 2 = 9 and C 2 : x - 3 2 + y - 4 2 = 16 , intersect at the points X and Y . Suppose that another circle C 3 : x - h 2 + y - k 2 = r 2 satisfies the following conditions: i

Accepted Answer

Given centre of C 1 , C 2 and C 3 are collinear hence 0 0 1 3 4 1 h k 1 = 0 3 k = 4 h …(i) ⇒ M N is diameter of C 3 M N = M C 1 + C 1 C 2 + C 2 N 2 r = r 1 + C 1 C 2 + r 2 ∵ M N = 2 r 2 r = 3 + 3 - 0 2 + 4 - 0 2 + 4 r = 6 …(ii) ⇒ Given C 3 touches C 1 at M So C 1 C 3 = r - 3 h 2 + k 2 = 9 …(iii) From (i) and (iii) h = ± 9 5 and k = ± 12 5 So centre of C 3 will be 9 5 , 12 5 ⇒ Now equation of common chord of C 1 and C 2 will be C 1 - C 2 = 0 6 x + 8 y = 18 Equation of line X Y is 3 x + 4 y = 9 …(

Question 6

Let the circles C 1 : x 2 + y 2 = 9 and C 2 : x - 3 2 + y - 4 2 = 16 , intersect at the points X and Y . Suppose that another circle C 3 : x - h 2 + y - k 2 = r 2 satisfies the following conditions: i

Accepted Answer

Given centre of C 1 , C 2 and C 3 are collinear hence 0 0 1 3 4 1 h k 1 = 0 3 k = 4 h …(i) ⇒ M N is diameter of C 3 M N = M C 1 + C 1 C 2 + C 2 N 2 r = r 1 + C 1 C 2 + r 2 ∵ M N = 2 r 2 r = 3 + 3 - 0 2 + 4 - 0 2 + 4 r = 6 …(ii) ⇒ Given C 3 touches C 1 at M So C 1 C 3 = r - 3 h 2 + k 2 = 9 …(iii) From (i) and (iii) h = ± 9 5 and k = ± 12 5 So centre of C 3 will be 9 5 , 12 5 ⇒ Now equation of common chord of C 1 and C 2 will be C 1 - C 2 = 0 6 x + 8 y = 18 Equation of line X Y is 3 x + 4 y = 9 …(

Question 7

A line y = m x + 1 intersects the circle x - 3 2 + y + 2 2 = 25 at the points P and Q . If the midpoint of the line segment P Q has x - coordinate - 3 5 , then which one of the following options is co

Accepted Answer

Equation of line P Q : y = m x + 1 Let R be the mid point of P Q , so its x -coordinate of midpoint = - 3 5 Hence, coordinates of midpoint R - 3 5 , - 3 m 5 + 1 Let centre of circle is O 3 , 2 Since, perpendicular dropped from centre upon any chord divides it in two equal parts. Hence, line P Q will be perpendicular to R O ⇒ m R Q ⋅ m R O = - 1 ⇒ m - 3 m 5 + 1 + 2 - 3 5 - 3 = - 1 ⇒ m - 3 m + 15 - 18 = - 1 ⇒ m 2 - 5 m + 6 = 0 ⇒ m = 2 , 3 ⇒ 2 ≤ m 4

Circle — Previous Year Questions & Solutions