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# Geometry

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A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The line is x = 4, and the equation of the circle is x^2 + y^2 = 25.

Jul 24, 2024

#1
+1655
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Let's create a system of equations to solve this question.

First, we know that the two points must be on the equation $$x^2 + y^2 = 25$$

We also know that x must be 4.

Thus, plugging in x=4 into the first equation, we can find y values. We have

$$4^2+y^2=25\\ y^2=25-16\\ y=\pm\sqrt9\\ y=3, y=-3$$

Thus, the two points A and B are $$(4,3), (4,-3)$$

Since they share an x value, the difference between the y values is the distance. We have

$$3-(-3)=6$$

Thus, 6 is our final answer.

Thanks! :)

Jul 25, 2024
edited by NotThatSmart  Jul 25, 2024

#1
+1655
+1

Let's create a system of equations to solve this question.

First, we know that the two points must be on the equation $$x^2 + y^2 = 25$$

We also know that x must be 4.

Thus, plugging in x=4 into the first equation, we can find y values. We have

$$4^2+y^2=25\\ y^2=25-16\\ y=\pm\sqrt9\\ y=3, y=-3$$

Thus, the two points A and B are $$(4,3), (4,-3)$$

Since they share an x value, the difference between the y values is the distance. We have

$$3-(-3)=6$$

Thus, 6 is our final answer.

Thanks! :)

NotThatSmart Jul 25, 2024
edited by NotThatSmart  Jul 25, 2024