MHB [ASK] A Line Intercepting A Circle

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A circle with center at (2, 1) intersects the line represented by the equation 3x + 4y + 5 = 0 at points A and B, with the distance AB measuring 8. The discussion reveals that the correct equation of the circle is not among the provided options, as none of them touch the line when checked with Desmos. To find the correct equation, one must calculate the distance from the circle's center to the line and use the properties of an isosceles triangle formed by the radius and the segment AB. The radius must be greater than 4, indicating that the radius squared exceeds 16. The thread concludes with a consensus that the initial options do not yield a valid circle equation.
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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. $$x^2+y^2-24x-2y-20=0$$
B. $$x^2+y^2-24x-2y-4=0$$
C. $$x^2+y^2-12x-2y-11=0$$
D. $$x^2+y^2-4x-2y+1=0$$
E. $$x^2+y^2-4x-2y+4=0$$

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.
 
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Monoxdifly said:
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. $$x^2+y^2-24x-2y-20=0$$
B. $$x^2+y^2-24x-2y-4=0$$
C. $$x^2+y^2-12x-2y-11=0$$
D. $$x^2+y^2-4x-2y+1=0$$
E. $$x^2+y^2-4x-2y+4=0$$

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
 
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.
 
$(x-2)^2 + (y-1)^2 = r^2$

$AB = 8 \implies r >4 \implies r^2 > 16$
 
Okay, thank you for all your answers. :)
 
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