Question

The equation of two diameters of a circle are x-2y+1=0 and x+y-2=0 and the length of chord intercepted on the straight line 3x+4y+8=0 by the circle is 6 units.find the equation of circle?

Answer 1

Given that the equations of two diameters of a circle are

`x-2y+1=0.......[1]`

and

`x+y-2=0.......[2]`

Subtracting [1] from [2] we get

`3y-3=0=>y=1`

Inserting `y=1` in [1]we get

`x-2*1+1=0=>x=1`

So the coordinates of point of intersection of two diameters is `(1,1)`. It is the coordinates of the center of the circle `O`.

The length of the perpendicular to the chord AB, `3x+4y+8=0` from the center of the circle will be `OC=(3*1+4*1+8)/sqrt(3^2+4^2) =3`

The Length of the chord `AB=6units`. `C` is the mid point of `AB`

So `AC=3 units`

Now `"radius"^2 =OA^2=OC^2+AC^2=3^2+3^2=18`

Hence equation of the circle is

`(x-1)^2+(y-1)^2=18`