Multiple choice

Find the equation to the circles which pass through the origin and cut off equal chords $a$ from the straight lines $y = x$ and $y = -x$.

  1. ${ x }^{ 2 }\pm \sqrt { 2 } ay+{ y }^{ 2 }=0$
  2. ${ -x }^{ 2 }\pm \sqrt { 2 } ay+{ y }^{ 2 }=0$
  3. ${ x }^{ 2 }\pm \sqrt { 2 } ay-{ y }^{ 2 }=0$

  4. None of the above
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A Correct answer
AI explanation

For a circle passing through the origin with equation x^2 + y^2 + 2gx + 2fy = 0, the centre is (-g, -f) and the radius squared is g^2 + f^2. The perpendicular distance from the centre to the lines y = x or y = -x is |-g + f| / sqrt(2). Using the relation radius^2 = distance^2 + (half chord length)^2, we write g^2 + f^2 = [(-g + f)^2 / 2] + (a/2)^2. Solving this gives the condition (g + f)^2 = a^2 / 2, meaning either g is 0 and f is plus or minus a / sqrt(2), or f is 0 and g is plus or minus a / sqrt(2). Substituting these values back results in the equations x^2 + y^2 plus or minus sqrt(2)ay = 0 and x^2 + y^2 plus or minus sqrt(2)ax = 0.

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