Multiple choice

If the locus of the middle point of the chords of contact of tangents to $x^{2}-y^{2}=a^{2}$ from points on the auxiliary circle is of the form $px^{4}+qy^{2}x^{2}+py^{4}-r(x^{2}+y^{2})=0$ then $r=$

  1. $a^{2}$
  2. $-a^{2}$
  3. $a$
  4. $-a$
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A Correct answer
Explanation

The chord of contact from (x1, y1) to x^2 - y^2 = a^2 is xx1 - yy1 = a^2. The midpoint of this chord is (h, k). The condition for the chord of contact to have a midpoint (h, k) is xh - yk = h^2 - k^2. Comparing coefficients leads to r = a^2.

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