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Difficulty: HardEquations and Graphs of Circles

In the standard (x,y)(x, y) coordinate plane, a circle is tangent to the xx-axis at the point (5,0)(5, 0). If the circle is also tangent to the line y=43xy = \frac{4}{3}x and its center lies in the first quadrant, what is the yy-coordinate of the center of the circle?

Answer: 2.5

Answer

The yy-coordinate of the center of the circle is 2.52.5.
A circle tangent to the xx-axis at (5,0)(5, 0) has a center along the vertical line x=5x = 5. Since the center is in the first quadrant, its coordinates can be represented as (5,k)(5, k) where k>0k > 0, and the radius is kk. The distance from the center (5,k)(5, k) to the line 4x−3y=04x - 3y = 0 must also equal the radius kk. Using the point-to-line distance formula, we write ∣4(5)−3k∣42+(−3)2=k\frac{|4(5) - 3k|}{\sqrt{4^2 + (-3)^2}} = k, which simplifies to ∣20−3k∣=5k|20 - 3k| = 5k. Since the center must lie in the first quadrant (k>0k > 0), we solve 20−3k=5k20 - 3k = 5k to find k=2.5k = 2.5. The alternative case 20−3k=−5k20 - 3k = -5k gives k=−10k = -10, which lies in the fourth quadrant and is thus excluded.

Step-by-Step Solution

1
Identify the coordinates of the center and the radius of the circle based on the xx-axis tangency.
Center: (5,k)(5, k) and Radius: r=kr = k (where k>0k > 0).
A circle tangent to the xx-axis at (5,0)(5, 0) has its center directly above or below this point on the line x=5x = 5. Since the center is in the first quadrant, its yy-coordinate kk must be positive, making the radius kk.
2
Use the distance from the center to the second tangent line to write an equation for kk.
∣4(5)−3k∣42+(−3)2=k\frac{|4(5) - 3k|}{\sqrt{4^2 + (-3)^2}} = k
The distance from the center (5,k)(5, k) to the tangent line 4x−3y=04x - 3y = 0 must equal the radius of the circle.
3
Simplify the equation and solve for kk.
∣20−3k∣=5k|20 - 3k| = 5k, leading to k=2.5k = 2.5 or k=−10k = -10.
Simplifying the denominator yields 55. Multiplying both sides by 55 gives the absolute value equation, which resolves to 20−3k=5k20 - 3k = 5k or 20−3k=−5k20 - 3k = -5k.
4
Select the valid solution using the quadrant constraint.
k=2.5k = 2.5
The center must lie in the first quadrant, which requires k>0k > 0. Thus, k=−10k = -10 is discarded, and the correct value is 2.52.5.

Key Concept

The relationship between a circle's center, its radius, and its tangent lines in the coordinate plane.
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Topics

Equations and Graphs of Circles

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