EXPLANATION.

Equation of tangent and equation of NORMAL to the giving curve to the.

⇒ x²/a² + y²/b² = 1.  at (a cosθ, B sinθ).

As we know that,

⇒ x²/a² + y²/b² = 1.

Differentiate w.r.t x, we get.

⇒ 2x/a² + 2y/b². (dy/dx) = 0.

⇒ 2y/b². (dy/dx) = - 2x/a².

⇒ dy/dx = (-2x/a²)/(2y/b²).

⇒ dy/dx = (-2x/a²) x (b²/2y).

⇒ dy/dx = -xb²/ya².

PUT the value of (a cosθ, b sinθ) in the equation, we get.

⇒ dy/dx = -(a cosθ) x b²/(b sinθ) x a².

⇒ dy/dx = (- bcosθ)/(a sinθ).

As we know that,

FORMULA of equation of tangent.

⇒ (y - y₁) = m(x - x₁).

Put the value in the equation, we get.

⇒ (y - b sinθ) = (- bcosθ)/(a sinθ) (x - a cosθ).

⇒ (a sinθ)(y - b sinθ) = (- bcosθ)(x - a cosθ).

⇒ (a sinθ)y - absin²θ = (- bcosθ)x + abcos²θ.

⇒ ay sinθ + bx cosθ = abcos²θ + absin²θ.

⇒ ay sinθ + bx cosθ = ab.

⇒ bx cosθ + ay sinθ = ab.

⇒ (bx cosθ)/ab + (ay sinθ)/ab = 1.

⇒ x/a cosθ + y/b sinθ = 1.

As we know that,

Formula of equation of normal.

⇒ (y - y₁) = -1/m(x - x₁).

⇒ (y - b sinθ) = (a sinθ)/(b cosθ) (x - a cosθ).

⇒ (b cosθ)(y - b sinθ) = (a sinθ)(x - a cosθ).

⇒ by cosθ - b²sinθ.cosθ = AX sinθ - a²sinθ.cosθ.

⇒ ax sinθ - by cosθ = (a² - b²) sinθ.cosθ.

⇒ (ax sinθ)/(sinθ.cosθ) - (by cosθ)/(sinθ.cosθ) = (a² - b²).

⇒ (ax)/cosθ - (by)/sinθ) = (a² - b²).

⇒ ax secθ - by cosecθ = (a² - b²).