Question

The equation of tangent to the curve $9x^2+16y^2=144$ which makes equal intercepts with coordinate axes, is

• A. $x+y=5$
• B. $x+y=16$
• C. $x+y=15$
• D. $Noneofthese$

Answer 1

The correct option is A $x+y=5$

Curve : $9x^2+16y^2=144$

$\Rightarrow \frac{x^2}{16}+\frac{y^2}{9}=1$

$\therefore$ Equation of tangent to the curve is

$\frac{x}{4}\cos \theta +\frac{y}{3}\sin \theta =1$

$\Rightarrow \frac{x}{\frac{4}{\cos \theta }}+\frac{y}{\frac{3}{\sin \theta }}=1$

$\because \frac{4}{\cos \theta }=\frac{3}{\sin \theta }$ (tangent makes equal intercepts with coordinates axes)

$\Rightarrow \tan \theta =\frac{3}{4}$

$\Rightarrow \theta ={\tan }^{-1}\left(\frac{3}{4}\right)$

$={37}^o\therefore \cos \theta =\frac{4}{5},\sin \theta =\frac{3}{5}$

of tangent $\Rightarrow \frac{x}{4}\times \frac{4}{5}+\frac{y}{3}\times \frac{3}{5}=1$

$\Rightarrow x+y=5$