The circle that is tangent to the x-axis at the point (8, 0) and passes through the point (0, -4)
A) $( x + 8 ) ^ { 2 } + ( y - 10 ) ^ { 2 } = - 10$
B) $( x + 10 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 100$
C) $( x - 8 ) ^ { 2 } + ( y + 10 ) ^ { 2 } = - 10$
D) $( x - 8 ) ^ { 2 } + ( y + 10 ) ^ { 2 } = 100$

Question

The circle that is tangent to the x-axis at the point (8, 0) and passes through the point (0, -4) 
A) $( x + 8 ) ^ { 2 } + ( y - 10 ) ^ { 2 } = - 10$
B) $( x + 10 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 100$
C) $( x - 8 ) ^ { 2 } + ( y + 10 ) ^ { 2 } = - 10$
D) $( x - 8 ) ^ { 2 } + ( y + 10 ) ^ { 2 } = 100$

Quizplus · Accepted Answer

The center of the circle is at (8, 0) + (0, radius) = (8, radius), because the circle is tangent to the x-axis at (8, 0). Since it also passes through (0, -4), the radius is the distance between (8, 0) and (0, -4), which is $\sqrt{(8-0)^2 + (0-(-4))^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}$. The center, therefore, is at (8, 0 - 4\sqrt{5}) = (8, -4\sqrt{5}). Since $-4\sqrt{5}$ is not an integer, none of the given centers in the options directly match the calculated center, indicating a mistake in the calculation of the center or radius. Correcting this, the radius should be calculated as the distance from (8, 0) to (0, -4), which is indeed $\sqrt{8^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80} = 8\sqrt{5}$, but the correct approach to find the center is to recognize that the circle is tangent to the x-axis at (8, 0), meaning the radius is perpendicular to the tangent point, making the radius equal to the y-coordinate's absolute value for points on the circle. The correct center is thus (8, 0) with a radius found by the distance to (0, -4), which is 8 units (not directly using the square root calculation). The error in the explanation above is in misinterpreting the center's location and the radius calculation. The correct radius is the distance from the center to any point on the circle, which should be calculated correctly to match the given options. Given the points and the correct understanding that the circle is tangent to the x-axis at (8, 0), the radius is half the distance in the y-direction between the tangent point and the point the circle passes through, which is incorrect as the radius should be directly calculated from the definition of a circle. The correct calculation involves recognizing the circle's properties based on the given points, leading to option D as the correct choice because it correctly represents a circle with a radius that fits the description of being tangent to the x-axis at (8, 0) and passing through (0, -4), with the correct formula for a circle being $(x - h)^2 + (y - k)^2 = r^2$, where (h, k) is the center and r is the radius. The mistake in the initial explanation involves incorrect assumptions about the center and radius, leading to confusion about the correct answer. The correct center is (8, 0), and by finding the radius correctly as the distance from the center to (0, -4), which is 8 units, and applying the formula for a circle, option D is identified as the correct equation of the circle based on the given conditions.

The Circle That Is Tangent to the X-Axis at the Point