Question

Statement $I$: The locus of the point, whose distance from the $x$-axis is twice its distance from the $y$-axis is $y^2=4x^2$
Statement $II$: The locus of the point $(\cot \theta +\cos \theta ,\cot \theta -\cos \theta )$ is $(x^2-y^2{)}^2=16xy$ .
Which of the above statements is/are true?

• A. Only $I$
• B. Only $II$
• C. Both $I$ and $II$
• D. Neither $I$ nor $II$

Answer 1

Answer: (C)

Both $I$ and $II$

Let the point be $(h,k)$.

According to the first statement, $|k|=2|h|$

$\Rightarrow k^2=4h^2$

i.e. $y^2=4x^2$

For the second statement,

$h=\cot \theta +\cos \theta$...(i)

$k=\cot \theta -\cos \theta$....(ii)

$\left(\frac{h+k}{2}\right)=\cot \theta$....adding eq (i) and (ii) .......(a)

$\left(\frac{h-k}{2}\right)=\cos \theta$...subtracting eq (i) and (ii).....(b)

${\left(\frac{h+k}{2}\right)}^2=\frac{{\cos }^2\theta }{si{n}^2\theta }=\frac{{\cos }^2\theta }{1-{\cos }^2\theta }$....substitute the value from eq (b)

$\Rightarrow {\left(\frac{h+k}{2}\right)}^2=\frac{(h-k{)}^2}{4-(h-k{)}^2}$

$\Rightarrow [4-(h-k{)}^2]\times (h+k{)}^2=4(h-k{)}^2$

$\Rightarrow 4[h^2+k^2+2hk-h^2-k^2+2hk]=(h^2-k^2{)}^2$

$\Rightarrow 16hk=(h^2-k^2{)}^2$

$\Rightarrow (h^2-k^2{)}^2=16hk$

$\Rightarrow (x^2-y^2{)}^2=16xy$