Question

$2x+y=0$ is the equation of a diameter of the circle which touches the lines $4x-3y+10=0$ and $4x-3y-30=0$ The center and radius of the circle are ?

• A. $(-2,1);4$
• B. $(1,-2);8$
• C. $(1,-2);4$
• D. $(1,-2);16$

Answer 1

The correct option is C $(1,-2);4$

Given $4x-3y+10=0$ and $4x-3y-30=0$ touches circle implies they are tangent.

Solving the line $2x+y=0$ and $4x-2y+10=0$

$x=-1$ and $y=2$ Point A

Solving the line $2x+y=0$ and $4x-3y-30=0$

$x=3;y=-6$ Point B

Distance between the parallel lines is length of diameter

$d=\frac{(C_1-C_2)}{\sqrt{(a^2+b^2)}}$

$d=\frac{(10-(-30)}{\sqrt{(16+9)}}$

$d=\frac{10+30}{5}$

$d=8$

$r=4$

O is midpoint of AB

$(x,y)=\frac{3-1}{2},\frac{2-6}{2}$ $=(1,-2)$