Question

The light from a lamp creates a shadow on a wall with a hyperbolic border. Find the equation of the border if the distance between the vertices is inches and the foci are inches from the vertices. Assume the center of the hyperbola is at the origin.

Answer 1

This question is incomplete, the complete question is;

The light from a lamp creates a shadow on a wall with a hyperbolic border. Find the equation of the border if the distance between the vertices is 18 inches and the foci are 4 inches from the vertices. Assume the center of the hyperbola is at the origin.

Answer:

the equation of the border is x²/81 - y²/88 = 1

Step-by-step explanation:

Given the data in the question;

distance between vertices = 2a = 18 in

so

a = 18/2 = 9 in

distance of foci from vertices = 4 in

hence distance between two foci = 2c = 18 + 4 + 4 =  26

so

c = 26/2 = 13 in

Now, from Pythagorean theorem

b = √( c² - a² )

we substitute

b = √( (13)² - (9)² ) = √( 169 - 81 ) = √88

we know center is at the origin, so

( h, k ) = ( 0, 0 )

h = 0

k = 0

Using equation of hyperbola

[ ( x-h )² / a² ] - [ ( y - k )² / b² = 1

we substitute

[ ( x-0 )² / 9² ] - [ ( y - 0 )² / (√88)² ] = 1

x²/81 - y²/88 = 1

Therefore, the equation of the border is x²/81 - y²/88 = 1