Multiple choice

The locus of the middle points of the chords of the circle ${x}^{2}+{y}^{2}=4$ which pass through $(3,4)$ is

  1. ${x}^{2}+{y}^{2}+4x+3y=0$
  2. ${x}^{2}+{y}^{2}-4x-3y=0$
  3. ${x}^{2}+{y}^{2}-3x-4y=0$
  4. ${x}^{2}+{y}^{2}+3x+4y=0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the midpoint be (h, k). The chord passing through (3, 4) has a midpoint (h, k). The vector from origin to (h, k) is perpendicular to the chord. The slope of the line from origin to (h, k) is k/h. The slope of the chord is (k-4)/(h-3). Since they are perpendicular, (k/h) * ((k-4)/(h-3)) = -1. k^2 - 4k = -h^2 + 3h => h^2 + k^2 - 3h - 4k = 0. Replacing (h, k) with (x, y) gives x^2 + y^2 - 3x - 4y = 0.

AI explanation

The locus of the midpoint of chords drawn from an external point (x1, y1) to a circle x squared plus y squared equals r squared is given by the equation T equals S1. Substituting the point (3, 4) into the formula gives x squared plus y squared minus 3x minus 4y equals 0. This matches the required locus equation.

Keep practicing — related questions