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Q.

Let a tangent drawn at any point on the ellipse x225+y216=1 cut the x - axis at Q. Let R be the  image of Q with respect  to y = x. If S is a circle with QR as it’s  diameter, then the fixed point through which the circle S  passes is

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a

(5,4) 
 

b

(4,5) 
 

c

(0,0) 
 

d

(0,5) 
 

answer is C.

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Detailed Solution

Given ellipse x225+y216=1 (a>b)

where a2=25, b2=16

Let P=(5 cosθ, 4 sinθ) be any point on ellipse

Now equation of tangent at P(θ) is x5 cosθ+y4sinθ=1

It cuts x-axis at Q=5cosθ,0

Given R=image of Q w.r to y=x

R=0,5cosθ

Now equation of circle having QR as diameter isx-x1(x-x2)+(y-y1)(y-y2)=0

x-5cosθx+(y)y-5cosθ=0 x2+y2-5xcosθ-5ycosθ=0

It passes through the point (0, 0)

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Let a tangent drawn at any point on the ellipse x225+y216=1 cut the x - axis at Q. Let R be the  image of Q with respect  to y = x. If S is a circle with QR as it’s  diameter, then the fixed point through which the circle S  passes is