Multiple choice

The coordinate that the chord $x\cos\alpha +y\sin\alpha -p=0$ of $x^2+y^2-a^2=0$ may subtend a right angle at the centre of the circle is?

  1. $a^2=2p^2$
  2. $p^2=2a^2$
  3. $a=2p$
  4. $p=2a$
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A Correct answer
Explanation

For a chord to subtend a right angle at the center, the distance from the center (0,0) to the line x*cos(alpha) + y*sin(alpha) = p must be a/sqrt(2). Thus, p = a/sqrt(2), which squares to p^2 = a^2/2 or a^2 = 2p^2.

AI explanation

The distance from the center (0,0) to the chord x cos a + y sin a = p is found using the perpendicular distance formula, giving d = p. Because the chord subtends a right angle at the center, the perpendicular from the center to the chord equals half the length of the chord. The half-length of the chord is also found using the Pythagorean theorem as sqrt(a^2 - p^2), and equating the two expressions for the half-length gives sqrt(a^2 - p^2) = p, which simplifies to a^2 = 2p^2.

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