Question

if x = a sec theta and y = b tan theta ,
then find the value of b²x²-a²y² ​

Answer 1

Given :

$x = a \secθ \\ y = b \: \tanθ$

Solution:

$\frac{x}{a} = \secθ.............(1) \\ \\ \frac{y}{b} = \tan θ..............(2)$

Using identity,

${ \secθ}^{2} - { \tan θ}^{2} = 1$

∴ from (1) and (2)

$( { \frac{x}{a} )}^{2} - ( { \frac{y}{b} )}^{2} = 1$

$\frac{ {x}^{2} }{ {a}^{2} } - \frac{ {y}^{2} }{ {b}^{2} } = 1$

By cross multiplying.

$\frac{ {b}^{2} {x}^{2} - {a}^{2} {y}^{2} }{ {a}^{2} {b}^{2} } = 1</strong><strong>\</strong><strong>\</strong><strong> </strong></p><p>\: \: \\</p><p><strong>$

Multiplying both sides by a^2b^2

b^2 x^2 - a^2 y^2= a^2 b^2

∴Value is a^2 b^2

Answer 2

Given,

• x = a sec θ
• y = b tan θ

To Find,

• The values of b²x² - a²y².

According to question,

Given, x = a sec θ

⇒ x/a = sec θ

Given, y = b tan θ

⇒ y/b = tan θ

We know that,

sec²θ - tan²θ = 1

[ Put the values ]

⇒ (x/a)² - (y/b)² = 1

⇒ x²/a² - y²/b² = 1

⇒ b²x² - a²y²/a².b² = 1

⇒ b²x² - a²y² = a².b²

Therefore,

The value of b²x² - a²y² is a².b².

For information :

Reciprocal Identities,

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ