Question

If the line $y=x+k$ touches the hyperbola $9x^2-16y^2=144$, then $k=$

• A. $7$
• B. $-7$
• C. $\pm \sqrt{7}$
• D. $\pm \sqrt{19}$

Answer 1

The correct option is C $\pm \sqrt{7}$

The given line is
$y=x+k$ .... (i)
Comparing with $y=mx+c$, we get
$m=1$
$c=k$
The given hyperbola is
$9x^2-16y^2=144$ .....(ii)
$\Rightarrow \frac{9x^2}{144}-\frac{16y^2}{144}=1$ [Dividing throughout by $144$]
$\Rightarrow \frac{x^2}{16}-\frac{y^2}{9}=1$
Comparing with $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, we get
$a^2=16$
and $b^2=9$
If line (i) touches the hyperbola (ii), then
$c^2=a^2{m}^2-b^2$
$\Rightarrow k^2=\left(16\right)(1{)}^2-9$
$\Rightarrow k^2=16-9$
$\Rightarrow k^2=7$
$\Rightarrow k=\pm \sqrt{7}$