# [ASK] A Line Intercepting A Circle

• MHB
Monoxdifly
MHB
A circle whose center is (2, 1) intercepts a line whose equation is 3x + 4y + 5 = 0 at point A and B. If the length of AB = 8, then the equation of the circle is ...
A. $$\displaystyle x^2+y^2-24x-2y-20=0$$
B. $$\displaystyle x^2+y^2-24x-2y-4=0$$
C. $$\displaystyle x^2+y^2-12x-2y-11=0$$
D. $$\displaystyle x^2+y^2-4x-2y+1=0$$
E. $$\displaystyle x^2+y^2-4x-2y+4=0$$

I don't know how to do it. Judging by the center of the circle, the answer must be either D or E. However, when I checked both of them with Desmos, neither circles even touches the line. How should I do it? Even if there's no right option, I would still like to know in case I encounter this kind of question again.

jonah1
Beer induced reaction follow.
A circle whose center is (2, 1) intercepts a line whose equation is 3x + 4y + 5 = 0 at point A and B. If the length of AB = 8, then the equation of the circle is ...
A. $$\displaystyle x^2+y^2-24x-2y-20=0$$
B. $$\displaystyle x^2+y^2-24x-2y-4=0$$
C. $$\displaystyle x^2+y^2-12x-2y-11=0$$
D. $$\displaystyle x^2+y^2-4x-2y+1=0$$
E. $$\displaystyle x^2+y^2-4x-2y+4=0$$

I don't know how to do it. Judging by the center of the circle, the answer must be either D or E. However, when I checked both of them with Desmos, neither circles even touches the line. How should I do it? Even if there's no right option, I would still like to know in case I encounter this kind of question again.
https://www.desmos.com/calculator/ngflni4a1s

Gold Member
MHB
Find the distance $h$ from the circle center $O$ to the line using this formula. Then you have an isosceles triangle with base $AB=8$ and height $h$. Find the equal legs of the triangle, which is the radius of the circle.

I don't see the correct answer in any of the variants. I believe the red circle on your sketch is the correct one.

skeeter
$(x-2)^2 + (y-1)^2 = r^2$

$AB = 8 \implies r >4 \implies r^2 > 16$

Monoxdifly
MHB