FIND EQUATIONS OF TANGENT DRAWN FROM THE POINT AWAY FROM THE CURVE

Problem 1 :

Find the equations of the tangents to the parabola y² = 5x from the point (5, 13). Also find the points of contact.

Solution :

The equation of the parabola is y² = 5x.

y² = 4ax

5x = 4ax

a = 5/4

Equation of the tangent  y = mx + a/m

y = mx + 5/4m --- (1)

Passes through the points (5, 13).

13 = 5m + 5/4m

13 = (20m² + 5)/4m

52m = 20m² + 5

20m² - 52m + 5 = 0

20m² - 2m - 50m + 5 = 0

2m(10m - 1) - 5(10m - 1) = 0

(2m - 5) (10m - 1) = 0

2m - 5 = 0 and 10m - 1 = 0

2m = 5 and 10m = 1 

m = 5/2 and m = 1/10

m = 5/2 substitute the equation (1).

m = 1/10 substitute the equation (1).

So, the points of contact are (1/5, 1), (125, 25).

Problem 2 :

Find the equations of the two tangents that can be drawn from the point (5, 2) to the ellipse 2x² + 7y² = 14.

Solution :

Given, 2x² + 7y² = 14

Dividing 14 on each sides.

Equation of tangent :

Problem 3 :

Find the equation of the two tangents that can be drawn

(i)  From the point (2, -3) to the parabola y² = 4x.

(ii)  From the point (1, 3) to the ellipse 4x² + 9y² = 36.

(iii)  From the point (1, 2) to the hyperbola 2x² - 3y² = 6.

Solution :

(i)  Given, y² = 4x

y² = 4ax

4x = 4ax

a = 1

Passes through the points (2, -3).

Equation of the tangent to the parabola will be of the form y = mx + 1/m.

-3 = 2m + 1/m

-3 = (2m² + 1)/m

-3m = 2m² + 1

2m² - 3m + 1 = 0

(ii)  Given, 4x² + 9y² = 36

Dividing 36 on each sides.

Equation of tangent :

(iii) Given, 2x² - 3y² = 6

Dividing 6 on each sides.

Equation of tangent : 

y - y₁ = m(x - x₁)

when m = 1

y - 2 = 1(x - 1)

y - 2 = x - 1

y - 2 - x + 1 = 0

y - x - 1 = 0

Dividing -1 on each sides.

x - y  + 1 = 0

when m = -3

y - 2 = -3(x - 1)

y - 2 = -3x + 3

y - 2 + 3x - 3 = 0

3x + y - 5 = 0