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Question

Column IColumn II(A)If real numbers x and y satisfy (x+5)2+(y12)2=81,then the minimum value of x2+y2 is(P)1(B)The line 3x+6y=k intersects the curve2x2+2xy+3y2=1 at point A and B.If the circle with AB as a diameter passesthrough the origin, then the value of k2 is(Q)2(C)If two perpendicular tangents can bedrawn from the origin to the circlex26x+y22py+17=0 , thenthe value of |p| is(R)4(D)If the circlesx2+y2+(3+sinβ)x+(2cosα)y=0 andx2+y2+(2cosα)x+2cy=0 touch each other, then the maximum value of c is(S)5(T)7(U)9
Which of the following is the only CORRECT combination?

A
(C)(R), (D)(Q)
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B
(C)(S), (D)(P)
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C
(C)(P), (D)(S)
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D
(C)(T), (D)(P)
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Solution

The correct option is B (C)(S), (D)(P)
(C)
Given:
x2+y26x2py+17=0(x3)2+(yp)2=(p28)
As, the tangents from origin to the given circle are perpendicular to each other,
so origin must lie on the director circle of the given circle.

Equation of director circle,
(x3)2+(yp)2=2(p28)
Putting (0,0) in the equation of the director circle,
9+p2=2p216
p2=25
p=±5
From the given options,
p=5
(C)(S)

(D)
x2+y2+(3+sinβ)x+(2cosα)y=0
x2+y2+(2cosα)x+2cy=0
Both circle touch each other at the origin, so tangent at origin will be same on both circles.
(3+sinβ)2x+(cosα)y=0 (1)
(cosα)x+cy=0 (2)

Comparing equation (1) and (2),
3+sinβ2cosα=cosαc
c=2cos2α3+sinβ
The maximum value of c occurs when cosα=1, sinβ=1.
cmax=1
(D)(P)

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