Question

if x is equal to a sec theta cos theta Y is equal to b sec theta sin theta and Z is equal to c tan theta show that x square Upon A square + Y square upon B square minus Z Square upon c square is equal to 1

Answer 1

hope this will help you .........

Answer 2

Answer:

Given: $x=a\,sec\theta\,cos\theta\:,\:y=b\,sec\theta\,sin\theta\:\:and\:\:z=c\,tan\theta$

To show: $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

Proof,

Consider

$x=a\,sec\theta\,cos\theta\\\frac{x}{a}=\frac{1}{cos\theta}\times cos\theta\\\\\frac{x}{a}=1\\\\ Squaring\,both\,sides,\,we\,get\\\\\frac{x^2}{a^2}=1 ...................... (1)$

$y=b\,sec\theta\,sin\theta\\\frac{y}{b}=\frac{1}{cos\theta}\times sin\theta\\\\\frac{y}{b}=tan\theta\\\\ Squaring\,both\,sides,\,we\,get\\\\\frac{y^2}{b^2}=tan^2\theta ...................... (2)$

$z=c\,tan\theta\\\frac{z}{c}=tan\theta\\\\\ Squaring\,both\,sides,\,we\,get\\\\\frac{z^2}{c^2}=tan^2\theta ...................... (3)$

Now consider,

$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}\\\\=\,1\,+\,tan^2\theta\,-\,tan^2\theta\:(from\,(1),(2),(3))\\\\=\,1$

Hence Proved.