Question

Elimate Φfrom the following X = 3 secΦ,y= 4tanΦ​

Answer 1

Step-by-step explanation:

Given,

$x = 3sec \theta$

$y = 4tan \theta$

To Do :-

Eliminate theta:-

Solution:-

As,

$x = 3sec \theta - - (1)$

$y = 4tan \theta - - (2)$

L.C.M of 3,4 = 12

Multiplying (1)with 4 and (2) with 3:-

Multiplying equation 1 with 4:-

$4 \times x = 4 \times 3sec \theta$

$4x = 12sec \theta$

Squaring on both sides :-

$(4x) ^{2} = (12sec \theta) ^{2}$

$\sf 16x^{2} = 144sec^{2}\theta$ --- (3)

Multiplying equation 2 with 3:-

$3 \times y = 3 \times 4tan \theta$

$3y = 12tan \theta$

Squaring on both sides :-

$\sf (3y)^{2} = (12tan\theta)^{2}$

$\sf 9y^{2} = 144tan^{2}\theta$---(4)

Equation '3' - Equation'4' :-

$16 {x}^{2} - 9 {y}^{2} = 144 {sec}^{2} \theta - 144 {tan}^{2} \theta$

$16 {x}^{2} - 9 {y}^{2} = 144( {sec}^{2} \theta - {tan}^{2} \theta)$

We know that :-

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

$16 {x}^{2} - 9 {y}^{2} = 144(1)$

$16 {x}^{2} - 9 {y}^{2} = 144$

Hence,

$16 {x}^{2} - 9 {y}^{2} = 144$