MAT Lab - Elipse & Line Question

[eyanveom]

I have a question on an assignment that's due at 11:55 tonight. Here is the question:

Compute the intersection of a line and an ellipse centered at (0,0). Ellipse equation is b²x² + a²y² = a²b²
where b is the minor axis and a is the major axis.

I am having trouble finding A, B, and C- somewhere down the line, I know the Quadratic Eq. is used to find x. I also know AX² + BX + C = 0 is used. I've combined the slope (y = mx + c) into the ellipse equation ==> b²x² + a² (mx+c)² = a² b². I then distributed the a², yielding b²x² + (a²mx + a²c)² = a²b². Here is where I'm stuck. I know that ultimately I am trying to find A, B, and C.

A= (a²mx + a²c)²??
B= ??
C= b²x² - a²b²?? I am SO lost and confused! lol

 

[Valkarie]

To compute the intersection of a line and an ellipse centered at (0,0), you need to substitute the equation of the line into the equation of the ellipse. The equation of the ellipse is b²x² + a²y² = a²b², and the equation of the line is y = mx + c. Substitute the equation of the line into the equation of the ellipse and simplify to get an equation in the form Ax² + Bx + C = 0. This will be a quadratic equation which can be solved for x using the quadratic formula. The coefficients A, B and C can be found as follows: A= a²(m² + 1) B= 2a²mc C= a²c² - a²b²