Multiple choice

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is

  1. $20\left(x^2 + y^2 \right) - 36x + 45y = 0 $
  2. $20\left(x^2 + y^2 \right) + 36x - 45y = 0 $
  3. $20\left(x^2 + y^2 \right) - 20x + 45y = 0 $
  4. $20\left(x^2 + y^2 \right) + 20x - 45y = 0 $
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A Correct answer
Explanation

The chord of contact of a point (h, k) with respect to x^2 + y^2 = r^2 is hx + ky = r^2. Given the point lies on 4x - 5y = 20, we have 4h - 5k = 20, or 4h = 5k + 20. The mid-point (x, y) of the chord of contact satisfies x = h(r^2)/(h^2 + k^2) and y = k(r^2)/(h^2 + k^2). Substituting r^2 = 9 and the relation between h and k leads to the resulting locus.

AI explanation

The chord of contact for a point (x1, y1) on the circle x^2 + y^2 = 9 is given by T = 0, yielding the equation xx1 + yy1 = 9. If (h, k) is the midpoint of this chord, applying the midpoint of a chord formula S1 = T, we get h^2 + k^2 - 9 = hh1 + kk1. Since the external point lies on the line 4x - 5y = 20, we have 4x1 - 5y1 = 20. Using these relationships, x1 equals (h^2 + k^2 - 9)9h divided by the substitution, leading to the elimination of x1 and y1, resulting in the locus equation 20(x^2 + y^2) - 36x + 45y = 0.

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