Find the equation of the circle which touches the lines `4x-3y+10=0a n d4x-3y-30=0` and whose centre lies on the line `2x+y=0.`

Answer

Radius =4 units; Centre C (1,-2)

Explanation

Lines `4x-3y+10=0` and `4x-3y-30=0` are two parallel tangents.
Diameter is distance between these two lines, which is `(|10-(-30)|)/(sqrt((4)^(2)+(-3)^(2)))=8` units.
`:. ` Radius `=` 4 units.
Centre lies on the line `2x+y=0`, so let the centre be `C(h,-2h)`.
Now, distance of C from any of tangent lines is 4.
`:. (|4h+6h+10|)/(sqrt((4)^(2)+(-3)^(2)))=4`
`:. |h+1|=2`
`:. h=1`
Therefore, centre is C(1,-2) (as centre lies between the given lines).