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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.

Jun 27, 2024

### Best Answer

#1
+1230
+1

We have a system of equations we need to solve.

We have the system

$$x=4\\ x^2+y^2=25$$

Plugging in x as 4 into the second equation, we have

$$16+y^2=25\\ y^2=9\\ y= \pm 3$$

Thus, we find there are two ordered pairs and solutions to the equation. We have that

$$(4, 3); (4, -3)$$

The distance between the two is just $$3-(-3) = 6$$ since they share the same x value.

Thus, 6 is our answer.

Thanks! :)

Jun 27, 2024

### 1+0 Answers

#1
+1230
+1
Best Answer

We have a system of equations we need to solve.

We have the system

$$x=4\\ x^2+y^2=25$$

Plugging in x as 4 into the second equation, we have

$$16+y^2=25\\ y^2=9\\ y= \pm 3$$

Thus, we find there are two ordered pairs and solutions to the equation. We have that

$$(4, 3); (4, -3)$$

The distance between the two is just $$3-(-3) = 6$$ since they share the same x value.

Thus, 6 is our answer.

Thanks! :)

NotThatSmart Jun 27, 2024