Question

eliminate theta from the equations ​

Answer 1

Step-by-step explanation:

We have,

$x \tan \theta + y \sec \theta = m \\ x \tan \theta - y \sec \theta = n$

Adding both equations,

$2x \tan\theta= m + n$

$\implies \tan\theta= \frac{ m + n}{2x}$

Subtracting the given 2nd equation from the 1st,

$2y \sec\theta = m - n$

$\implies\sec\theta = \frac{m - n}{2y}$

Now, we know that,

$\sec^{2} \theta - \tan^{2} \theta = 1$

$\implies (\frac{m - n}{2y} )^{2} - ( \frac{m + n}{2x} )^{2} = 1$

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

$\implies (m - n)^{2} x ^{2} - (m + n)^{2}y ^{2} = 4 {x}^{2} {y}^{2}$