Determine the equations of the lines that are tangent to the ellipse x^2+4y^2=16 at the point x=2

Answer 1

If x = 2 then 2² + 4y² = 16; y² = 3 and y = ±√3. So we have 2 tangent points at x = 2: (2, √3) and (2, - √3).

Now we differentiate implicitly equation of ellipse: 2x + 8yy' = 0; y' = - x/(4y).

If tangent point is (2, √3) slope of tangent line y' = - 2/(4√3) = - √3/6. and equation of tangent line:

y - √3 = - √3/6(x - 2) or y = - √3x/6 + 4√3/3.

If tangent point is (2, - √3) slope of tangent line y' = - 2/(- 4√3) = √3/6. and equation of tangent line:

y + √3 = √3/6(x - 2) or y = √3x/4 - 4√3/3