Relationship Between an Ellipse and a Line

For the ellipse and the line , the discriminant of the quadratic equation formed by eliminating describes their relationship:

1. The line intersects the ellipse at two distinct points.
2. The line is tangent to the ellipse, touching at exactly one point.
3. The line does not intersect the ellipse.

Equation of the Tangent Line to an Ellipse

1. When the slope is given
   The equation of a line with slope tangent to the ellipse is:
2. For any given slope, there are two tangent lines to the ellipse.
   
   
3. When the coordinates of the point of tangency are given
   The equation of the tangent line at the point on the ellipse is:

Equation of a Tangent Line from a Point Outside the Ellipse

To find the equation of the tangent line from a point outside the ellipse , there are two methods:

1. [Method 1] Assume the coordinates of the point of tangency are . Use the tangent line equation , and apply the condition that the line passes through . Then, solve for and , knowing that lies on the ellipse.
   
   
2. [Method 2] Form a system of equations by combining the equation of the line passing through with slope , and the ellipse equation. Then, solve for by ensuring the discriminant of the quadratic equation is zero, which indicates tangency.