EQUATION OF TANGENT THAT IS PARALLEL OR PERPENDICULAR TO THE LINE

Find the equations of the tangents.

Problem 1 :

To the parabola y² = 6x, parallel to 3x - 2y + 5 = 0.

Solution :

y² = 6x

y² = 4ax

6x = 4ax

6/4 = a

3/2 = a

Equation of tangent y = mx + a/m

Slope of the tangent = Slope of the line

= -(a/b)

= -(3/-2)

m = 3/2

So, equation of tangents is 3x - 2y + 2 = 0.

Problem 2 :

To the parabola y² = 16x, perpendicular to the line 3x - y + 8 = 0.

Solution :

y² = 16x

y²  = 4ax

16x = 4ax

16/4 = a

4 = a

Equation of tangent y = mx + a/m

Slope of the tangent = Slope of the line

= b/a

m = -1/3

So, equation of tangents is x + 3y + 36 = 0.

Problem 3 :

Solution :

Slope of the tangent = Slope of the line

= b/a

m = 1/1

m = 1

So, equation of tangents is x - y + 5 = 0.

Problem 4 :

To the hyperbola 4x² - y² = 64, which are parallel to 10x - 3y + 9 = 0.

Solution :

4x² - y² = 64

Equation of tangent y = mx + a/m

Slope of the tangent = Slope of the line

= -(a/b)

= -(10/-3)

m = 10/3

So, equation of tangents is 10x - 3y + 32 = 0.