Equation of parabola :
y^{2} = 4ax
Condition for tangent :
c = a/m
Point of contact :
Equation of tangent :
Tangent of Ellipse
Equation of ellipse :
Condition for tangency :
c^{2} = a^{2} m^{2} + b^{2}
Point of contact :
Equation of tangent :
Equation of hyperbola :
Condition for tangency :
c^{2} = a^{2} m^{2} - b^{2}
Point of contact :
Equation of tangent :
Problem 1 :
Prove that the line 5x + 12y = 9 touches the hyperbola x^{2} - 9y^{2} = 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
a^{2} = 9 a = 3 |
b^{2} = 1 b = 1 |
c^{2} = a^{2}m^{2} - b^{2}
∴ point of contact is (5, -4/3).
Problem 2 :
Show that the line x - y + 4 = 0 is a tangent to the ellipse x^{2} + 3y^{2} = 12. Find the co-ordinates of the point of contact.
Solution :
x^{2} + 3y^{2} = 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
c^{2} = a^{2}m^{2} + b^{2}
(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 16x^{2} - 9y^{2} = 144 are the roots of x^{2} - (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 16x^{2} - 9y^{2} = 144
Dividing 144 on each sides.
a^{2} = 9 a = 3 |
b^{2} = 16 b = 4 |
Given, line is y = mx + 2√5 .
The condition is c^{2} = a^{2}m^{2} - b^{2}.
(2√5)^{2} = 3^{2}m^{2} - 4^{2}
20 = 9m^{2} - 16
9m^{2} = 20 + 16
9m^{2} = 36
m^{2} = 36/9
m^{2} = 4
m = ±2
Let a = 2 and b = -2
x^{2} - (a + b)x - 4 = 0
x^{2} - (sum of roots)x + products of roots = 0
a + b = 2 - 2 = 0
So, option (3) is correct.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM