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Difficulty: HardNonlinear Systems of Equations

In the xyxy-plane, the graph of the linear equation 3x+4y=k3x + 4y = k, where kk is a positive constant, is tangent to the circle with equation x2+y2−2x−4y=20x^2 + y^2 - 2x - 4y = 20. What is the value of kk?

Answer: 36

Answer

The value of the constant kk is 3636.
To find the value of kk where the line 3x+4y=k3x + 4y = k is tangent to the circle, we first rewrite the circle's equation x2+y2−2x−4y=20x^2 + y^2 - 2x - 4y = 20 in standard form by completing the square: (x2−2x+1)+(y2−4y+4)=20+1+4(x^2 - 2x + 1) + (y^2 - 4y + 4) = 20 + 1 + 4, which simplifies to (x−1)2+(y−2)2=25(x - 1)^2 + (y - 2)^2 = 25. This is a circle centered at (1,2)(1, 2) with a radius of 55. A line is tangent to a circle if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle. Using the distance formula ∣Ax0+By0−C∣A2+B2\frac{|Ax_0 + By_0 - C|}{\sqrt{A^2 + B^2}} for the point (1,2)(1, 2) and the line 3x+4y−k=03x + 4y - k = 0, we set up the equation: ∣3(1)+4(2)−k∣32+42=5\frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = 5. This simplifies to ∣11−k∣5=5\frac{|11 - k|}{5} = 5, or ∣11−k∣=25|11 - k| = 25. Solving the absolute value equation gives 11−k=25⇒k=−1411 - k = 25 \Rightarrow k = -14 and 11−k=−25⇒k=3611 - k = -25 \Rightarrow k = 36. Since the problem specifies that kk is a positive constant, the correct answer is 3636.

Step-by-Step Solution

1
Rewrite the circle's equation by completing the square for both variables.
(x−1)2+(y−2)2=25(x - 1)^2 + (y - 2)^2 = 25
To identify the center and radius of the circle.
2
Identify the center and radius from the standard form of the circle's equation.
Center is (1,2)(1, 2) and radius is 55.
The standard form of a circle is (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2.
3
Apply the condition for tangency: the perpendicular distance from the center (1,2)(1, 2) to the line 3x+4y−k=03x + 4y - k = 0 must equal the radius 55.
∣3(1)+4(2)−k∣32+42=5\frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = 5
A line is tangent to a circle if and only if the distance from the center to the line equals the radius.
4
Simplify the distance equation and solve the resulting absolute value equation for kk.
∣11−k∣=25|11 - k| = 25, yielding k=−14k = -14 or k=36k = 36.
To find all mathematically possible values of the constant kk.
5
Select the positive value for kk as specified by the problem constraints.
k=36k = 36
The problem states that kk is a positive constant.

Key Concept

The relationship between a line and a circle in a nonlinear system, specifically using the distance from the center to a tangent line to solve for an unknown constant.
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Topics

Nonlinear Systems of Equations

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