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Question
If two distinct chords, drawn from the point(p, q) on the circle x2+y2=px+qy(where pq≠0) are bisected by the x-axis, then
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Solution
The correct option is D >
Let PQ be a chord of the given circle passing through P(p, q) and the coordinates of Q be (x, y)
Since PQ is bisected by the x-axis, the mid-point of PQ lies on the x axis which gives y = - q
Now Q lies on the circle x2+y2−px−qy=0
⇒x2+q2−px+q2=0⇒x2−px+2q2=0→(i)
which gives two values of x and hence the coordinates of two points Q and R (say), so that the chords PQ and PR are bisected by x-axis. If the chords PQ and PR are distinct, the roots of (i) are real distinct.
⇒ the discriminant p2−8q2>0 ⇒p2>8q2
>Let PQ be a chord of the given circle passing through P(p, q) and the coordinates of Q be (x, y)
Since PQ is bisected by the x-axis, the mid-point of PQ lies on the x axis which gives y = - q
Now Q lies on the circle x2+y2−px−qy=0
⇒x2+q2−px+q2=0⇒x2−px+2q2=0→(i)
which gives two values of x and hence the coordinates of two points Q and R (say), so that the chords PQ and PR are bisected by x-axis. If the chords PQ and PR are distinct, the roots of (i) are real distinct.
⇒ the discriminant p2−8q2>0 ⇒p2>8q2
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