Question

If the line \[y=2x+c\] be a tangent to the ellipse \[\frac{x^2}{8}+\frac{y^2}{4}=1\], then c = ______.

Options

• \[\text{土4}\]

• \[\text{土6}\]

• \[\text{土1}\]

• \[\text{土8}\]

Answer 1

If the line \[y=2x+c\] be a tangent to the ellipse \[\frac{x^2}{8}+\frac{y^2}{4}=1\], then c = \[\text{土6}\].

Explanation:

The ellipse is: \[\frac{x^2}{8}+\frac{y^2}{4}=1\]

So \[a^2=8,b^2=4.\]

The condition for the line \[y=mx+c\] to be a tangent to the ellipse \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\] is:

                             \[c^2=a^2m^2+b^2\]

Here m = 2, so:

                           \[c^2=8\times(2)^2+4\]

                           \[c^2=8\times4+4\]

                           \[c^2=32+4=36\]

                           \[c=\pm6\]