40000 live tutoring sessions served!
At Home Tuition, Inc BBB Business Review
Click here to Live Chat
1-888-592-9928

Hyperbolas

Definition of a hyperbola

A hyperbola is the locus of a point that moves in the plane in such a way that the ratio of its distance from a fixed point (focus) in the same plane to its distance from a fixed line (directrix) in the plane is always constant and greater than unity.

As in the previous cases, the constant ratio is the eccentricity e and e > 1 for every hyperbola.

The fixed point is the focus S, the fixed line is the directrix ZZ1 and if P(x,y) is any point on the hyperbola, then

                 SP/PM = e

        ⇒    SP= e.PM

 

Example 1:

 

Find the equation of the hyperbola whose focus is (1, 2), directrix is the line x+y+1=0 and eccentricity is e = 3/ 2.

Solution:

Let P(x,y) be a point on the hyperbola.

Let PM = perpendicular from P onto the directrix

 
 
By cross-multiplying
 
 

is the equation of the required hyperbola.

 

Try these questions

I) Find the equation of the hyperbola whose

 

i.

Focus is (-1, 1) Directrix x - y + 3= 0 and eccentricity is 3

ii.

Focus is (2, -1) Directrix is 2x + 3y = 1 and eccentricity is 2

iii.

Focus is (a, 0) Directrix is 2x - y + a = 1 and eccentricity is 4/3

iv.

Focus is (2, 2) Directrix is x + y = 9 and eccentricity is 3/2
 

Answers
i.

Solution:

 
 
 
is the equation of the required hyperbola.
 
ii.

Solution:
 

Given focus =S = (2, -1)

Directrix is 2x + 3y = 1 => 2x + 3y - 1 = 0

                 e = 2

Let P(x,y) be any point on the hyperbola

Let PM = perpendicular from P onto the directrix

 
 
Cross multiplying
 
 
is the equation of the required hyperbola.
 
iii.
 Solution:
 

Given focus =S = (a, 0)

Directrix is 2x - y + a = 0

                 e = 4/3

Let P(x,y) be any point on the hyperbola

Let PM = perpendicular from P onto the directrix

 
 
is the equation of the required hyperbola.
 
iv.

Solution:
 

Given focus =S= (2, 2)

Directrix is x + y = 9 => x + y - 9 = 0

                 e = 3/2

Let P(x,y) be any point on the hyperbola

Let PM = perpendicular from P onto the directrix

 
 
  is the equation of the required hyperbola.
   

Related Topics