Question

Find the equation of tangent and normal to the curve given by x – 7 cos t andy = 2 sin t, t ∈ R at any point on the curve

Answer 1

x = 7 cos t and y = 2 sin t, t ∈ R

Differentiating w.r.t. ‘t’,

`("d"x)/"dt"` = – 7 sin t and `("d"y)/"dt"` = 2 cos t

Slope of the tangent ‘m’

`("d"y)/("d"x) = (("d"y)/("dt"))/(("d"x)/("d"t"))`

= `(2 cot"t")/(- 7 sin "t")`

Any point on the curve is (7 Cos t, 2 sin t)

Equation of tangent is y – y₁ = m (x – x₁)

y – 2 sint = `- (2cot"t")/(7sin"t")` (x – 7 cos t)

7y sin t – 14 sin²t = – 2x cos t + 14 cos²t

2x cos t + 7 y sin t – 14(sin²t + cos²t) = 0

2x cos t + 7y sin t – 14 = 0

Now slope of normal is `- 1/3 = (7sin"t")/(2cos"t")`

Equation of normal is y – y₁ = `- 1/"m"` (x – x₁)

y – 2 sin t = `(7sin"t")/(2cos"t")` (x – 7 cos t)

2y cos t – 4 sin t cos t = 7x sin t – 49 sin t cos t 7x sin t – 2y cos t – 45 sin t cos t = 0