Question

Find the value(s) of $k$ so that the line $2x+y+k=0$ may touch the hyperbola $3x^2-y^2=3$

Answer 1

The given line is $y=-2x-k$ ........ $\left(i\right)$
The given hyperbola is $3x^2-y^2=3$ ........ $\left(ii\right)$

For the point of intersection of $\left(i\right)$ and $\left(ii\right)$ substituting the value of $y$ from $\left(i\right)$ in equation $\left(ii\right)$, we get
$3x^2-(-1{)}^2(2x+k{)}^2=3$
$3x^2-(4x^2+4kx+k^2)=3$
$3x^2-4x^2-4kx-k^2-3=0$
$x^2+4kx+k^2+3=0$ ....... $\left(iii\right)$

Now the line $\left(i\right)$ will touch the hyperbola if equation $\left(iii\right)$ has equal roots.
Discriminant $=0$
i.e., $(4k{)}^2-4(1\left)\right(k^2+3)=0$
$16k^2-4k^2-12=0$
$12k^2-12=0$
$k^2=1$
$k=\pm 1$