Multiple choice

From the points on the circle $x^2+y^2=a^2$, tangents are drawn to the hyperbola $x^2-y^2=a^2$; the locus of the middle points of the chords of contact is

  1. $\displaystyle\frac{x^2-y^2}{a^2x^2+y^2}=a^2$
  2. ${(x^2-y^2)}^2=a^2(x^2+y^2)$
  3. ${(x^2+y^2)}^2=a^2(x^2-y^2)$
  4. $x^2+y^2=a^2(x^2-y^2)$
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B Correct answer
Explanation

The chord of contact of tangents from (x1, y1) to the hyperbola x^2 - y^2 = a^2 is xx1 - yy1 = a^2. The midpoint of this chord is (h, k), so the equation of the chord is xh - yk = h^2 - k^2. Comparing these, we find h/x1 = -k/y1 = a^2 / (h^2 - k^2). Since (x1, y1) lies on the circle x^2 + y^2 = a^2, substituting these values leads to the locus (x^2 - y^2)^2 = a^2(x^2 + y^2).

AI explanation

Let a point on the circle be (a cos theta, a sin theta). The equation of the chord of contact to the hyperbola x^2 - y^2 = a^2 from this point is x cos theta - y sin theta = a. If (h, k) is the midpoint of this chord of contact, we use the formula T = S1, which gives hx - ky = h^2 - k^2. Comparing the two equations, we get cos theta = h/a and sin theta = k/a. Using the identity cos^2 theta + sin^2 theta = 1, we have (h/a)^2 + (k/a)^2 = 1, or h^2 + k^2 = a^2. Substituting a^2 = h^2 + k^2 into the midpoint relation h^2 - k^2 = a^2, the locus is (x^2 - y^2)^2 = a^2(x^2 + y^2).

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