Question

The value of $\lambda$ does the line $y=x+\lambda$ touches the ellipse $9x^2+16y^2=144$ is/are

• A. $\pm 2\sqrt{2}$
• B. $2\pm \sqrt{3}$
• C. $\pm 5$
• D. $5\pm \sqrt{2}$

Answer 1

Answer: (C)

$\because$ Equation of ellipse is $9x^2+16y^2=144$ or
$\frac{x^2}{16}+\frac{y^2}{9}=1$
Comparing this with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ then we
get $a^2=16$ and $b^2=9$ and comparing the line
$y=x+\lambda$ with $y=mx+c$
$\therefore m=1$ and $c=\lambda$
If the line $y=x+\lambda$ touches the ellipse $9x^2$
$+16y^2=144$, then $c^2=a^2{m}^2+b^2$
$\Rightarrow {\lambda }^2=16\times 1^2+9\Rightarrow {\lambda }^2=25$
$\therefore \lambda =\pm 5$