Multiple choice

From the origin, chords are drawn to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}$ -2y $=0$. The locus of the middle point of these chords is

  1. $\mathrm{x}^{2}+\mathrm{y}^{2} -\mathrm{y} =0$
  2. $\mathrm{x}^{2}+\mathrm{y}^{2} -2\mathrm{x} =0$
  3. $\mathrm{x}^{2}+\mathrm{y}^{2} -\mathrm{x} =0$
  4. $\mathrm{x}^{2}+\mathrm{y}^{2} -\mathrm{x}-\mathrm{y} =

    0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the midpoint of the chord be (h, k). The equation of the chord with a given midpoint is T = S1, which gives x*h + y*k - (y + k) = h^2 + k^2 - 2k. Since the chord passes through the origin (0,0), substituting x = 0 and y = 0 yields h^2 + k^2 - k = 0, which represents the locus x^2 + y^2 - y = 0.

AI explanation

Let the midpoint of the chords be (h, k). Since the chords originate from the origin (0, 0), the midpoint is given by ((h+0)/2, (k+0)/2) = (h, k), which simplifies to (h, k) = (h, k) and implies the other end of the chord is (2h, 2k). Because (2h, 2k) must lie on the circle x^2 + y^2 - 2y = 0, we substitute the coordinates to get 4h^2 + 4k^2 - 4k = 0, which simplifies to h^2 + k^2 - k = 0. Replacing h with x and k with y gives the locus x^2 + y^2 - y = 0.

Keep practicing — related questions