Multiple choice

The chord of contact of the pair of tangents drawn from any point on the line $3x + y = 5$ to the circle $\displaystyle x^{2}+y^{2}=4 $ passes through

  1. $\displaystyle \left ( \frac{1}{5},\frac{2}{7} \right ) $
  2. $\displaystyle \left ( \frac{12}{7},\frac{4}{7} \right ) $
  3. $\displaystyle \left ( \frac{12}{5},\frac{4}{5} \right ) $
  4. $(1, 1)$
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C Correct answer
Explanation

The chord of contact of a point (h, k) with respect to x^2 + y^2 = 4 is hx + ky = 4. Since (h, k) lies on 3x + y = 5, we have k = 5 - 3h. Substituting this into the chord equation gives hx + (5 - 3h)y = 4, or 5y - 4 + h(x - 3y) = 0. This line passes through the intersection of 5y - 4 = 0 and x - 3y = 0, which is y = 4/5 and x = 3(4/5) = 12/5.

AI explanation

Let the point on the line 3x + y = 5 be (h, k). The equation of the chord of contact of tangents drawn from this point to the circle x^2 + y^2 = 4 is given by T = 0, which is hx + ky = 4. Since the point lies on the line, we have 3h + k = 5. We need to find a fixed point (x, y) through which hx + ky = 4 passes for all such h and k. Rearranging the linear condition 3h + k = 5, we can align it with hx + ky = 4 to find the ratios x/3 = y/1 = 4/5. Solving this gives the fixed point as (12/5, 4/5).

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