Question

The Locus of the point $(\tan \theta +\sin \theta ,\tan \theta -\sin \theta )$ is $(x^m-y^m{)}^m=axy$. Find $a\times m$

Answer 1

locus of the point $(tan\theta +sin\theta ,tan\theta -sin\theta )$

$x=tan\theta +sin\theta$

$y=tan\theta -sin\theta$

$tan\theta =\frac{x+y}{2},sin\theta =\frac{x-y}{2}$

$cos\theta =\frac{x-y}{x+y}$

$si{n}^2\theta +co{s}^2\theta =1$

$\frac{(x-y{)}^2}{4}+\frac{(x-y{)}^2}{(x+y{)}^2}=1$

$(x-y{)}^2[4+(x+y{)}^2]=4(x+y{)}^2$

$\Rightarrow (x+y{)}^2(x-y{)}^2=4\left[\right(x+y{)}^2-(x-y{)}^2]$

$\Rightarrow (x+y{)}^2=4[4xy]$

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

$a=16,m=2$

$axm=16\times 2=32$