Question

find the values of c if a striaght line y = 2x + c, is tangent to the ellipse:
(x^2)/9 + (y^2)/4 = 1

Answer 1

Using the top half of the ellipse, we have
y = 2/3 √(9-x^2)
For the ellipse to intersect the line y=2x+c, we need
2/3 √(9-x^2) = 2x+c
4/9 (9-x^2) = (2x+c)^2
40/9 x^2 + 4cx + c^2-4
For the ellipse to be tangent to the line (only one solution), we need the discriminant to be zero.
16c^2 - 4(40/9)(c^2-4) = 0
So, c = 2√10

See the graph at
https://www.wolframalpha.com/input/?i=plot+(x%5E2)%2F9+%2B+(y%5E2)%2F4+%3D+1,+y%3D2x%2B2%E2%88%9A10

There is a similar point where c = -2√10