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Difficulty: MediumSystems of Linear and Non-Linear Equations

A circle in the standard (x,y)(x, y) coordinate plane is defined by the equation (x−2)2+y2=20(x - 2)^2 + y^2 = 20, and a line is defined by the equation y=2x−4y = 2x - 4. If the circle and the line intersect at two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), what is the value of x1+x2+y1+y2x_1 + x_2 + y_1 + y_2?

Answer: 4

Answer

The value of x1+x2+y1+y2x_1 + x_2 + y_1 + y_2 is 4.
Substituting the equation of the line into the circle's equation gives (x−2)2+(2x−4)2=20(x-2)^2 + (2x-4)^2 = 20. Since (2x−4)=2(x−2)(2x-4) = 2(x-2), we can write this as (x−2)2+4(x−2)2=20(x-2)^2 + 4(x-2)^2 = 20, which simplifies to 5(x−2)2=205(x-2)^2 = 20, and further to (x−2)2=4(x-2)^2 = 4. Solving for xx yields x−2=2x - 2 = 2 or x−2=−2x - 2 = -2, so x=4x = 4 or x=0x = 0. Substituting these values into the linear equation gives the points (4,4)(4, 4) and (0,−4)(0, -4). The sum of all coordinates is 4+0+4+(−4)=44 + 0 + 4 + (-4) = 4.

Step-by-Step Solution

1
Substitute the linear equation into the circle equation.
(x−2)2+(2x−4)2=20(x - 2)^2 + (2x - 4)^2 = 20
To find the coordinates of the intersection points by reducing the system to a single variable equation.
2
Simplify the equation using algebraic substitution.
5(x−2)2=205(x - 2)^2 = 20
Factoring 2 out of the term (2x−4)(2x-4) yields 2(x−2)2(x-2). Squaring it results in 4(x−2)24(x-2)^2. Combining this with the first term (x−2)2(x-2)^2 simplifies the expression.
3
Solve for the two possible x-coordinates.
x=0x = 0 and x=4x = 4
Dividing both sides by 5 yields (x−2)2=4(x-2)^2 = 4, which means x−2=±2x-2 = \pm 2.
4
Substitute the x-coordinates back into the linear equation to find the corresponding y-coordinates.
For x=0x = 0, y=−4y = -4, giving the point (0,−4)(0, -4). For x=4x = 4, y=4y = 4, giving the point (4,4)(4, 4).
To determine the coordinates of both intersection points.
5
Calculate the sum of all coordinates: x1+x2+y1+y2x_1 + x_2 + y_1 + y_2.
0+4+(−4)+4=40 + 4 + (-4) + 4 = 4
To find the final value requested by the question.

Key Concept

Solving systems of linear and quadratic (circular) equations by substitution and factoring.
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Topics

Systems of Linear and Non-Linear Equations

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