Question

Find the equations of the circles which pass through the origin and cut off equal chords of $\sqrt{2}$ units from the lines y = x and y = -x.

Answer 1

Figure shows four such circles.

angle between $y=x$ and $y=-x$ is $\frac{\pi }{2}$
$\therefore$ Angle between OB and OA $=\frac{\pi }{2}$
Hence, AB, BC, CD and AD are diameter of circles,
$\angle BOQ=\frac{\pi }{4}sin\angle BOQ=\frac{BQ}{OB}sin\frac{\pi }{4}=\frac{BQ}{\sqrt{2}}\frac{1}{\sqrt{2}}=\frac{BQ}{\sqrt{2}}BQ=1$
Radius of circles = 1 = OQ.
Coordinates of B is (1, 1)
Similarly, coordinates of (-1, 1), C(1, -1), D(-1, -1)
Equation of circle with diameter AB is
$(x+1)(x-1)+(y-1)(y-1)=0x^2-1+y^2-2y+1=0x^2+Y^2,-2y=0$
Equation of circle with diameter BC is
$(x-1)(x-1)+(y-1)(y+1)=0x^2-2x+1+y^2-1=0x^2+Y^2-2x=0$
Equation of circle with diameter CD is
$(x+1)(x-1)+(y-1)(y+1)=0x^2-1+y^2+2y+1=0x^2+y^2+2y=0$
Equation of circle with diameter AD is
$(x+1)(x+1)+(y-1)(y+1)=0x^2+2x+1+y^2-1=0x^2+y^2+2x=0$