# CONDITION OF TANGENCY FOR PARABOLA ELLIPSE AND HYPERBOLA

## Tangent of Parabola

Equation of parabola :

y2 = 4ax

Condition for tangent :

c = a/m

Point of contact :

Equation of tangent :

Tangent of Ellipse

Equation of ellipse :

Condition for tangency :

c2 = a2 m2 + b2

Point of contact :

Equation of tangent :

## Tangent of Hyperbola

Equation of hyperbola :

Condition for tangency :

c2 = a2 m2 - b2

Point of contact :

Equation of tangent :

Problem 1 :

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

 a2 = 9a = 3 b2 = 1b = 1

c2 = a2m2 - b2

∴ point of contact is (5, -4/3).

Problem 2 :

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).

Problem 3 :

The values of m for which y = mx + 2 root 5 touches the hyperbola 16x2 - 9y2 = 144 are the roots of x2 - (a + b)x - 4 = 0, then the value of (a + b) is

(1)  2  (2)  4  (3)  0 (4)  -2

Solution :

Equation of the hyperbola is 16x2 - 9y2 = 144

Dividing 144 on each sides.

 a2 = 9a = 3 b2 = 16b = 4

Given, line is y = mx + 2√5 .

The condition is c2 = a2m2 - b2.

(2√5)2 = 32m2 - 42

20 = 9m2 - 16

9m2 = 20 + 16

9m2 = 36

m2 = 36/9

m2 = 4

m = ±2

Let a = 2 and b = -2

x2 - (a + b)x - 4 = 0

x2 - (sum of roots)x + products of roots = 0

a + b = 2 - 2 = 0

So, option (3) is correct.

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