ORTHOGONAL BETWEEN TWO CURVES

Problem 1 :

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

Solution :

3x² = 12

x² = 4

x = 2 and -2

When x = 2, y = 1 + 2³ ==> 9

When x = -2, y = 1 + (-2)³ ==> -7

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

Equation of tangent :

(2, 9) and slope = 12

y - y₁ = m(x - x₁)

y - 9 = 12(x - 2)

y - 9 = 12x - 24

12x - y = -9 + 24

12x - y = 15

(-2, -7) and slope = 12

y - y₁ = m(x - x₁)

y + 7 = 12(x + 2)

y + 7 = 12x + 24

12x - y = 7 - 24

12x - y = -17

Problem 2 :

Show that the two curves x² - y² = r² and xy = c² where c, r are constants cut orthogonally.

Solution :

x² - y² = r²

2x - 2y(dy/dx) = 0

2y(dy/dx) = 2x

(dy/dx) = 2x/2y

(dy/dx) = x/y -----(1)

xy = c²

x(dy/dx) + y(1) = 0

x(dy/dx) = -y

dy/dx = -y/x -----(2)

(1) x (2)

= (x/y) ( -y/x)

= -1

The product of the slopes is equal to -1. So, the curves are orthogonal.