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Question

Two chords are drawn from the point P(h,k) on the circles x2+y2hxky=0. If the y-axis divides both the chords in the ratio 1:3, then which of the following is/are correct?

A
9k2>16h2
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B
k2>48h2
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C
9k216h2
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D
4k2=h2
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Solution

The correct option is B k2>48h2
Case-1: Ratio =1:3

(0,a)=(h+3x14,k+3y14)
0=h+3x14, a=k+3y14
x1=h3, y1=4ak3

Since, (x1,y1) lies on the circle.
x21+y21hx1ky1=0
h29+(4ak)29+h23+k24ak3=0
4a25ak+(h2+k2)=0

Now, D>0 for two different chords.
25k216(h2+k2)>0
9k2>16h2

Case-2: Ratio =3:1

(0,a)=(3h+x14,3k+y14)
0=3h+x14, a=3k+y14
x1=3h, y1=4a3k

9h2+(4a3k)2+3h2+3k24ak=0
4a27ak+3(h2+k2)=0

Now, D>0 for two different chords.
49k248(h2+k2)>0
k2>48h2

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