Multiple choice

The tangents at the extremities of each of two chords drawn from the point $(2, 2)$ to the circle $\displaystyle x^{2}+y^{2}-2x-4y+1=0 $ meet at points A and B, respectively. The equation of the line AB is

  1. $x = 5$
  2. $x + y = 4$
  3. $2x - y = 2$
  4. $y = 2$
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A Correct answer
Explanation

The chord of contact from a point (x1, y1) to a circle x^2+y^2+2gx+2fy+c=0 is xx1 + yy1 + g(x+x1) + f(y+y1) + c = 0. For (2,2) and the circle, the chord equation is 2x + 2y - 1(x+2) - 2(y+2) + 1 = 0, which simplifies to x - 5 = 0, or x = 5.

AI explanation

The equation of the circle is x^2 + y^2 - 2x - 4y + 1 = 0, so its centre is (1, 2) and radius is 2. The given point (2, 2) lies on the circle. The equation of the tangent at (2, 2) is x(2) + y(2) - x(1+2) - y(2+2) + 1 = 2^2 + 2^2 - 2*2 - 4*2 + 1, which simplifies to x = 5. The locus of the intersection of tangents at the ends of chords passing through a fixed point on the circle is simply the tangent at that point itself, so AB must be x = 5.

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