Question

find the equation of circle which touches the lines 4x-3y+10=0and 4x-3y-30=0 and whose centre lies on the line 2x+y=0​

Answer 1

Answer:

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

The given tangents are parallel and therefore the distance between them

$2r = \frac{10 + 30}{5} = 8 \\ r = 4$

Now the diameter 2x + y = 0 cuts the given line in the points P (-1, 2) and Q (3, -6) whose middle point is therefore the centre (1, - 2).

Hence its equation is

${(x - 1)}^{2} + {(y + 2y)}^{2} = {4}^{2} = 16$