Question

If the line `y=2x+c` touches the ellipse `x^2/4+y^2/3=1`, find the value of `c`?

Answer 1

`c=sqrt19 or c=-sqrt19`

Explanation:

The line `y=2x+c` touches the ellipse , that means there is only one solution of the equations `y=2x+c` and `x^2/4+y^2/3=1`
Now substituting the value of `y` in the equation of ellipse we get
`x^2/4 +(2x+c)^2/3=1`
`=>3x^2+16x^2+4c^2+16cx=12`
`=> 19x^2+16cx+4c^2-12=0`
Now as this equation has only one solution
`:.`Discriminant must be equal to zero
`:.(16c)^2-4×19×(4c^2-12)=0`
`:.256c^2-304c^2+912=0`
`=>48c^2=912`
`:.c=sqrt19 or c=-sqrt19`

graph{(3x^2+4y^2-12)(2x+sqrt19-y)(2x-sqrt19-y)=0 [-5, 5, -2.5, 2.5]}