# [ASK] A Line Intercepting A Circle

• MHB
• Monoxdifly
In summary, the answer to this question is that the distance from the center of the circle to the line is $h$, and the radius of the circle is $r$.
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.

Beer induced reaction follow.
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. $$\displaystyle x^2+y^2-24x-2y-20=0$$
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

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.

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

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

## 1. What is a line intercepting a circle?

A line intercepting a circle is a line that intersects a circle at two points. These points are called the points of intersection and are located on the circumference of the circle.

## 2. How can you determine the points of intersection between a line and a circle?

The points of intersection between a line and a circle can be determined by solving the equations of the line and the circle simultaneously. This can be done by substituting the equation of the line into the equation of the circle and solving for the values of x and y.

## 3. What is the equation for a line intercepting a circle?

The equation for a line intercepting a circle is y = mx + b, where m is the slope of the line and b is the y-intercept. This equation can be used to determine the points of intersection between the line and the circle.

## 4. Can a line intercept a circle at more than two points?

No, a line can only intercept a circle at two points. This is because a line and a circle can only intersect at a maximum of two points. If a line intersects a circle at more than two points, it would no longer be a line but a curved shape.

## 5. How is the distance between the points of intersection calculated?

The distance between the points of intersection can be calculated using the distance formula, which is d = √[(x2 - x1)² + (y2 - y1)²]. In this formula, (x1, y1) and (x2, y2) are the coordinates of the two points of intersection.

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