Multiple choice

Find the equation of the circle which passes through the origin and the cuts off chords of the lengths $4$ and $6$ on the positive side of $x-axis$ and $y-axis$ respectively.

  1. $x^{2}+y^{2}-4x-6y=0$
  2. $x^{2}-y^{2}+4x+6y=0$
  3. $-x^{2}+y^{2}+4x-6y=0$

  4. None of these
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A Correct answer
Explanation

Circle passes through (0,0). Chords on axes mean the circle intersects axes at (4,0) and (0,6). The equation of a circle passing through (0,0), (a,0), and (0,b) is x^2 + y^2 - ax - by = 0. Here a=4, b=6, so x^2 + y^2 - 4x - 6y = 0.

AI explanation

A circle passing through the origin takes the form x^2+y^2+2gx+2fy=0, giving x-intercepts at 0 and -2g, and y-intercepts at 0 and -2f. Since the positive x-axis chord length is 4, -2g=4 so g=-2, and since the positive y-axis chord length is 6, -2f=6 so f=-3. Substituting g and f into the general equation yields x^2+y^2-4x-6y=0.

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