Question

Find an equation of the tangent line at each given point on the curve. x = 2 − 3 cos θ, y = 3 + 2 sin θ

Answer 1

x = 2 - 3cos∅
(x -2) = -3cos∅
(2 - x)² = 9cos²∅ [ take square both sides,]
(2-x)²/9 = cos²∅ ----(1)

similarly
y = 3 + 2sin∅
(3 - y)²/4 = sin²∅ ----(2)

we know,
sin²∅ + cos²∅ = 1
from equations (1) and (2)
(2 -x)²/9 + (3 - y)²/4 = 1
hence,
(x -2)²/(3)² + (y-3)²/(2)² = 1 equation of ellipse .

now, slope of tangent = dy/dx
y = 3 + 2sin∅
dy/d∅ = 2cos∅
again,
x = 2 - 3cos∅
dx/d∅ = 3sin∅

hence, dy/dx = 2cot∅/3
so, general equation of tangent of ellipse 3(y -3 - 2sin∅)= 2cot∅(x - 2 + 3cos∅)