Find the equations of the circles which pass through the origin and cut off chords of length `a` from each of the lines `y=x and y=-x`

Answer

`x^(2)+y^(2)+- sqrt(2)ax=0` and `x^(2)+y^(2)+- sqrt(2)ay=0`

Explanation

In the figure, length of chords OA and AB that cut off by the circles is a.
`:. A-= ((a)/(sqrt(2)),(a)/(sqrt(2)))` and `B-= ((a)/(sqrt(2)),-(a)/(sqrt(2)))`, which are end points of diameters.
Therefore, the equation of circle is
`(x-(a)/(sqrt(2)))(x-(a)/(sqrt(2)))+(y-(a)/(sqrt(2)))(y+(a)/(sqrt(2)))=0`
or `x^(2)+y^(2)-sqrt(2)ax=0`
Circles with AC and BD as diametere are given by `x^(2)+y^(2)+- sqrt(2) ay=0`