If the tangents drawn from the origin to the circle x2 + y2 − 2px �

1.	If the tangents drawn from the origin to the circle x2 + y2 − 2px − 2qy - q2 = (q ≠ 0) are at right angles, then show that p2 = q2.
Answer» We know that y-axis touches the circle x² + y² + 2gx + 2fy + c = 0 if f² = c. For the circle mentioned in this problem also, we have q² = f² = c = q². Therefore, y-axis touches the circle. Since the tangents drawn from the origin are at right angles, the other tangent touches the x-axis. Therefore, p² = q² Aliter: Since the tangents drawn from (0, 0) are at right angles, origin must lie on the director circle of the given circle Therefore, (0, 0) lies on the circle (x − p)² + (y − q)² + 2p².

If the tangents drawn from the origin to the circle x2 + y2 − 2px − 2qy - q2 = (q ≠ 0) are at right angles, then show that p2 = q2.

Answer»

We know that y-axis touches the circle x² + y² + 2gx + 2fy + c = 0 if f² = c. For the circle mentioned in this problem also, we have q² = f² = c = q². Therefore, y-axis touches the circle. Since the tangents drawn from the origin are at right angles, the other tangent touches the x-axis. Therefore,

p² = q²

Aliter: Since the tangents drawn from (0, 0) are at right angles, origin must lie on the director circle of the given circle Therefore, (0, 0) lies on the circle (x − p)² + (y − q)² + 2p².

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