Multiple choice

Through a point $\left( h,k \right)$ secants are drawn to the circle $\left( { x }^{ 2 }+{ y }^{ 2 } \right) ={ r }^{ 2 }$. The mid-points of the corresponding chords describe the curve

  1. ${ x }^{ 2 }+{ y }^{ 2 }+hx+ky=0$
  2. ${ x }^{ 2 }+{ y }^{ 2 }-hx-ky=0$
  3. ${ x }^{ 2 }+{ y }^{ 2 }-2hx-2ky=0$
  4. ${ x }^{ 2 }+{ y }^{ 2 }+2hx+2ky=0$
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B Correct answer
Explanation

The midpoint (x, y) of a chord of the circle x^2 + y^2 = r^2 that passes through (h, k) satisfies the condition that the vector from the origin to the midpoint is perpendicular to the chord. This leads to the equation x(x-h) + y(y-k) = 0, which simplifies to x^2 + y^2 - hx - ky = 0.

AI explanation

Let the midpoint of the chord be denoted as (a, b), which serves as the foot of the perpendicular from the origin (0,0) to the line passing through (h, k). The slope of the line joining (0,0) and (a, b) is b/a, making the slope of the chord -a/b, allowing us to write the equation of the chord as bx - ay = b h - a k. Since (a, b) lies on this chord, we get a b - b h = -b k + b^2, which simplifies to a^2 + b^2 = a h + b k. Replacing the variables a and b with x and y yields the curve x^2 + y^2 - hx - ky = 0.

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