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
