Solve each problem. Find the equation of the line passing through the points of intersection of the graphs of x^2 + y^2 = 20 and x^2 - y = 0.

Verified step by step guidance

1. Step 1: Identify the equations of the two graphs. The first equation is a circle: \(x^2 + y^2 = 20\). The second equation is a parabola: \(x^2 - y = 0\).
2. Step 2: Solve the second equation for \(y\) in terms of \(x\). This gives \(y = x^2\).
3. Step 3: Substitute \(y = x^2\) from the second equation into the first equation \(x^2 + y^2 = 20\). This results in \(x^2 + (x^2)^2 = 20\).
4. Step 4: Simplify the equation from Step 3 to find the values of \(x\). This becomes \(x^2 + x^4 = 20\), which can be rearranged to \(x^4 + x^2 - 20 = 0\).
5. Step 5: Solve the quadratic equation in terms of \(x^2\) to find the values of \(x\), and then use these \(x\) values to find the corresponding \(y\) values using \(y = x^2\). These \((x, y)\) pairs are the points of intersection. Use these points to find the equation of the line passing through them.

Key Concepts

Circle Equation

The equation x^2 + y^2 = 20 represents a circle centered at the origin (0,0) with a radius of √20. Understanding the properties of circles, including their standard form and how to derive points on the circle, is essential for finding intersection points with other graphs.

Parabola Equation

The equation x^2 - y = 0 can be rewritten as y = x^2, which describes a parabola that opens upwards with its vertex at the origin. Recognizing the shape and characteristics of parabolas helps in determining where they intersect with other curves, such as circles.

Finding Intersection Points

To find the intersection points of the two graphs, one must solve the system of equations formed by the circle and the parabola. This typically involves substituting one equation into the other and solving for the variables, which yields the coordinates of the points where the two graphs meet.