[ASK] A Line Intercepting a Circle

[Monox D. I-Fly]

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. [tex]x^2+y^2-24x-2y-20=0[/tex]

B. [tex]x^2+y^2-24x-2y-4=0[/tex]

C. [tex]x^2+y^2-12x-2y-11=0[/tex]

D. [tex]x^2+y^2-4x-2y+1=0[/tex]

E. [tex]x^2+y^2-4x-2y+4=0[/tex]


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.

[Guest]

Let the center of the circle be denoted by [tex]C[/tex]. Now, we drop a perpendicular from [tex]C[/tex] onto [tex]AB[/tex]. Let the point of intersection be [tex]D[/tex]. Since this also bisects the chord, [tex]AD=BD[/tex]. Now, [tex]CA=\sqrt{{CD}^2+{AD}^2}[/tex] and this will give us the radius of the circle. To find [tex]CD[/tex], find the distance between [tex]C[/tex] and the line. I hope this helps you solve it.