Multiple choice

The chord of contact of pair of tangents drawn from a point $P$ to the circle $x^{2}+y^{2}=1$ passes through a fixed point $\left(1/2,1/4\right)$. Then $P$lies

  1. On the line $x+2y=4$
  2. On the line $2x+y=4$
  3. On the line $x-y=4$

  4. None of the these
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B Correct answer
Explanation

The chord of contact of P(x1, y1) to x^2 + y^2 = 1 is xx1 + yy1 = 1. It passes through (1/2, 1/4), so (1/2)x1 + (1/4)y1 = 1. Multiplying by 4 gives 2x1 + y1 = 4. Thus P(x1, y1) lies on the line 2x + y = 4.

AI explanation

Let the coordinates of point P be (h, k). The equation of the chord of contact from P to the circle x squared plus y squared equals 1 is hx + ky = 1. Since this chord of contact passes through the fixed point (1/2, 1/4), we substitute these coordinates to get h multiplied by 1/2 plus k multiplied by 1/4 equals 1. Multiplying the entire equation by 4 gives 2h + k = 4, showing that P lies on the line 2x + y = 4.

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