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In the $xy$ -plane, a circle of radius $4$ with center on the positive $x$ -axis is tangent to the $y$ -axis at the origin, and a circle with radius $10$ with center on the positive $y$ -axis is tangent to the $x$ -axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

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Evaluate two-variable equations: word problems

Full solution

Q. In the

xy

-plane, a circle of radius

4

with center on the positive

x

-axis is tangent to the

y

-axis at the origin, and a circle with radius

10

with center on the positive

y

-axis is tangent to the

x

-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

Identify Centers: Let's identify the centers of the two circles. The first circle has a radius of $4$ and is tangent to the y-axis at the origin, so its center is at $(4,0)$ . The second circle has a radius of $10$ and is tangent to the x-axis at the origin, so its center is at $(0,10)$ .
Find Slope: The points of intersection between the two circles lie on the line that passes through both centers, which are $(4,0)$ and $(0,10)$ . To find the slope of this line, we use the slope formula: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$ .
Calculate Slope: Plugging in the coordinates of the centers into the slope formula, we get: $\text{slope} = \frac{10 - 0}{0 - 4} = \frac{10}{-4} = -\frac{5}{2}$ .
Final Answer: The slope of the line passing through the two points at which the circles intersect is $-\frac{5}{2}$ . This is the final answer.

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