SOLUTION: state the position of each lie with respect to the ellipse x^2/9 + y^2/36 = 1 a.)y=2x+2 b.)y=-2x-9 c.)2x+5y+3=0 its differenet from the first one that i already ask

The way to solve this problem is as follows:


1. Express y via x from the given linear equation.

   (In your case it is just done in a.) and b.) ).


2. Substitute this expression into the equation for ellipse.

   You will get a quadratic equation.


3. If this quadratic equation has two real roots, then the straight line intersects the ellipse in two points.

   If this quadratic equation has one real root, then the straight line is tangent to the ellipse. 

   If this quadratic equation has no real roots, then the straight line has no common points with the ellipse. 
   I.e. the straight line is outside the ellipse.


4. Therefore, when you got the quadratic equation, calculate and check its discriminant.

   If the discriminant is positive, then the quadratic equation has two real roots. 
   Hence, the straight line intersects the ellipse in two points.

   If the discriminant is zero, then the quadratic equation has one real root. 
   Hence, the straight line is tangent to the ellipse. 

   If the discriminant is negative, then the quadratic equation has no real roots. 
   Hence, the straight line has no common points with the ellipse. I.e. the straight line is outside the ellipse.