Length of Intercept Made by a Circle on a Straight Line
Two chords ar...
Question
Two chords are drawn from the point P(h,k) on the circles x2+y2−hx−ky=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
9k2≥16h2
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D
4k2=h2
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Solution
The correct option is Bk2>48h2 Case-1: Ratio =1:3 (0,a)=(h+3x14,k+3y14) ⇒0=h+3x14,a=k+3y14 ⇒x1=−h3,y1=4a−k3
Since, (x1,y1) lies on the circle. ⇒x21+y21−hx1−ky1=0 ⇒h29+(4a−k)29+h23+k2−4ak3=0 ⇒4a2−5ak+(h2+k2)=0
Now, D>0 for two different chords. 25k2−16(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=4a−3k
⇒9h2+(4a−3k)2+3h2+3k2−4ak=0 ⇒4a2−7ak+3(h2+k2)=0
Now, D>0 for two different chords. 49k2−48(h2+k2)>0 ⇒k2>48h2