FIND POINT OF CONTACT OF TANGENT AND PARABOLA ELLIPSE AND HYPERBOLA

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 :

Prove that the line 5x + 12y = 9 touches the hyperbola x² - 9y² = 9 and find its point of contact. 

Solution :

5x + 12y = 9

12y = 9 - 5x

y = (9 - 5x)/12

y = mx + c

y = -5x/12 + 9/12

m = -5/12, c =  9/12 

c² = a²m² - b²

∴ point of contact is (5, -4/3).

Problem 3 :

Show that the line x - y + 4 = 0 is a tangent to the ellipse x² + 3y² = 12. Find the co-ordinates of the point of contact.

Solution :

x² + 3y² = 12

Dividing 12 on each sides.

Equation of the line is x - y + 4 = 0.

y = x + 4

y = mx  + c

where m = 1, c = 4

c² = a²m² + b²

(4)² = (1²)(1) + 4

16 = 12 + 4

16 = 16

∴ point of contact is (-3, 1).