Question

Tangent lines are drawn to the ellipse x^2 + 3y^2 = 12 at points (3,-1) and (-3,-1) on the ellipse.

1.  Find the equation of each tangent line. Write the answer in slope-intercept form.

2. Find the point at which the tangent lines intersect.

Answer 1

Differentiate the equation to find the tangent:

2x+6ydy/dx=0. At (3,-1) this becomes 6-6dy/dx=0 so dy/dx=1 and y+1=x-3, y=x-4 is the equation of the tangent line. At (-3,-1) this becomes -6-6dy/dx=0 and dy/dx=-1 and y+1=-(x+3), y=-x-4 is the equation of the tangent line.

The tangent lines intercept one another when x-4=-x-4, so x=0 and y=-4. The intersection is at (0,-4).