Q:

By solving the equations as a system, find the points common to the line with equation x-y=6 and the circle with equation x²+y²=26.Graph the line and the circle to show those points.

Accepted Solution

A:
Answer:The solutions to the system are the points (5, -1) and (1, -5)Step-by-step explanation:Hi there!We have the following system of equations:x - y = 6x² + y² = 26The points common to the line and the circle are those (x,y) values that satisfy both equations. So let´s take the first equation and solve it for x:x - y = 6add y to both sides of the equationx = 6 + yNow, let´s replace the x in the second equation:x² + y² = 26(6 + y)² + y² = 26(6 + y)(6 + y) + y² = 2636 + 12y + y² + y² = 26subtract 26 to both sides of the equation10 + 12y + 2y² = 0Solve the quadratic equation using the quadratic formula:a = 2b = 12c = 10[-b ± √(b² - 4ac)] / 2aThe solutions to the quadratic equation are y = -5 and y = -1Let´s calculate the x value:x = 6 + yFor y = -5x = 6 - 5 = 1For y = -1x = 6 - 1 = 5Then, the solutions to the system and the points at which the line and the circle intersect are (5, -1) and (1, -5). Please, see the attached figure to corroborate this graphically.Have a nice day!