CONDITION OF TANGENCY FOR PARABOLA ELLIPSE AND HYPERBOLA

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 

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.

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.