Question

The line $12x\cos \theta +5y\sin \theta =60$ is tangent to which of the following curves?

• A. $25x^2+12y^2=3600$
• B. $144x^2+25y^2=3600$
• C. $x^2+y^2=169$
• D. $x^2+y^2=60$

Answer 1

The correct option is B $144x^2+25y^2=3600$

$12x\cos \theta +5y\sin \theta =60\Rightarrow x\left(\frac{\cos \theta }{5}\right)+y\left(\frac{\sin \theta }{12}\right)=1\cdots \left(i\right)$
We know that the tangent to
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
At point $(a\cos \theta ,b\sin \theta )$ is given by
$\frac{x\cos \theta }{a}+\frac{y\sin \theta }{b}=1\cdots \left(ii\right)$
Comparing $\left(i\right)$ and $\left(ii\right)$, we get
$a=5$ and $b=12$
Hence, the curve is
$\frac{x^2}{25}+\frac{y^2}{144}=1\Rightarrow 144x^2+25y^2=3600$