Question

Find the equations of the tangents to the curve y = 1 + x³ for which the tangent is orthogonal with the line x + 12y = 12.

Answer 1

y = 1 + x³

dy/dx = 3x²

Substituting x₁ values in the curve when x₁ = 2, y₁ = 9; when x₁ = -2, y₁ = -1 

So the points are (2, 9) and (-2, -7) 

To find the equations of tangents: 

Tangents are orthogonal to x + 12y = 12 

So equations of tangents will be of the form 12x – y = k 

The tangent passes through (2, 9) ⇒ 24 – 9 = k ⇒ k = 15. 

∴ Equation of tangent is 12x – y = 15 

The tangent passes through (-2, -7) ⇒ 12 (-2) + 7 = k ⇒ -17 

So equation of tangent is 12x – y = -17