State whether the statements are true or false.

Equation of the line passing through the point (a cos³θ, a sin³θ) and perpendicular to the line x sec θ + y cosec θ = a is x cos θ – y sin θ = a sin 2θ.

Let the equation of line y = mx + c …(i)

So, slope of the above equation is ‘m’

Given equation of line is x sec θ + y cosec θ = a

⇒ y cosecθ = a – x secθ

Since the above equation is in y = mx + b form

So, slope of the equation is

Given that eq. (i) is perpendicular to x sec θ + y cosec θ = a

⇒ m × m’ = -1

Putting the value of m in eq. (i), we get

⇒ y secθ = x cosecθ + c (secθ)

⇒ x cosecθ – y secθ = - c secθ

⇒ x cosecθ – y secθ = k …(ii)

[Let k = - c secθ]

If eq. (ii) passes through the point (a cos³θ, a sin³θ)

⇒ (a cos³θ) cosecθ – (a sin³θ) secθ = k

⇒ a[(cos²θ – sin²θ)(cos²θ + sin²θ)] = xcosθ – ysinθ

[∵(a⁴ – b⁴) = (a² – b²)(a² + b²)]

⇒ a[(cos²θ – sin²θ)(1)] = xcosθ – ysinθ

[∵cos²θ + sin²θ = 1]

⇒ a[cos 2θ] = xcosθ – ysinθ

[∵cos²θ – sin²θ = cos 2θ]

⇒ xcosθ – ysinθ = a cos 2θ

Hence, the given statement is FALSE