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A circle centered at the origin is tangent to line x-2y+15=0 What is the area of the circle?

 Dec 2, 2023
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The distance from the center of the circle to the line is equal to the radius of the circle. To find the radius, we need to find the perpendicular distance from the origin to the line.

 

The equation of the line in slope-intercept form is y=21​x+215​. The slope of the line is 21​, and its negative reciprocal is −2. Therefore, the equation of the perpendicular line that passes through the origin is y=−2x. Setting the two equations equal to each other, we get −2x=21​x+215​. Solving for x, we get that x=−5.

 

Substituting this value of x back into the equation of the line, we get that y=5, so the origin is (0,5). Therefore, the radius of the circle is 5, so the area of the circle is π⋅52=25π​ square units.

 Dec 2, 2023

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