Answer each question. A line passes through the points of intersection of the graphs of y = x^2 and x^2 + y^2 = 90. What is the equation of this line?

Verified step by step guidance

1. Identify the points of intersection of the graphs y = \frac{3x^2}{50} and x^2 + y^2 = 136.
2. Substitute y = \frac{3x^2}{50} into the equation x^2 + y^2 = 136 to find the x-coordinates of the intersection points.
3. Solve the resulting equation for x to find the x-coordinates of the intersection points.
4. Substitute the x-coordinates back into y = \frac{3x^2}{50} to find the corresponding y-coordinates.
5. Use the two points of intersection to find the equation of the line passing through them using the point-slope form of a line.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c. In this context, the function y = x^2 represents a parabola that opens upwards, with its vertex at the origin (0,0). Understanding the properties of parabolas, such as their vertex, axis of symmetry, and direction of opening, is essential for analyzing their intersections with other curves.

Circle Equation

The equation x^2 + y^2 = r^2 represents a circle centered at the origin with radius r. In this case, the equation x^2 + y^2 = 90 describes a circle with a radius of √90. Recognizing the geometric properties of circles, including their symmetry and how they interact with other shapes, is crucial for determining points of intersection with other graphs, such as parabolas.

Finding Points of Intersection

Finding points of intersection involves solving a system of equations where two graphs meet. This typically requires substituting one equation into another to find common solutions. In this scenario, substituting y = x^2 into the circle's equation allows us to find the x-coordinates of the intersection points, which can then be used to derive the equation of the line that passes through these points.