Question

The value(s) of $\lambda$ for which the line $y=x+\lambda$ touches the ellipse
$9x^2+16y^2=144$ is/are:

• A. $5$
• B. $-5$
• C. $10$
• D. $-10$

Answer 1

Answer: (A), (B)

$-5$

$\mathrm{On}\mathrm{comparing}y=x+\lambda ,\mathrm{with}y=\mathrm{mx}+c$,$\mathrm{we}\mathrm{get}:$
$m=1\mathrm{and}c=\lambda \mathrm{and}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{ellipse}\mathrm{is}$ $9x^2+16y^2=144$ or $\frac{x^2}{16}+\frac{y^2}{9}=1\mathrm{On}\mathrm{comparing}\mathrm{this}\mathrm{equation}\mathrm{with}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\mathrm{we}\mathrm{get}:a=4\mathrm{and}b=3\mathrm{If}\mathrm{the}\mathrm{line}y=\mathrm{mx}+c\mathrm{touches}\mathrm{the}\mathrm{ellipse}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\mathrm{then}{c}^2=a^2{m}^2+b^2\Rightarrow {\lambda }^2=16+9=25\Rightarrow \lambda =\pm 5$