Question

Find the equation of the hyperbola whose asymptotes are 3x=+-5y and the vertices are (+-5,0).​

Answer 1

Given

3x + 5y and 3x - 5y are asymptotes to a hyperbola,say H.

To Find

Equation of hyperbola.

Concept

The product of equation of asymptotes is called A. The equation of the hyperbola is called H. The main concept is that A and H differs by a constant ,say k. That is,

$\boxed{\blue{A\:=\:H\:+\:K}}$

After that,to find the constant k ,we will put the co-ordinate of vertices in the equation of k because vertices lies on hyperbola.

Calculation

Let us first calculate A.

A = (3x + 5y) (3x - 5y)

=> A = (3x)² - (5y)²

=> A = 9x² - 25y²

Thus, 9x²- 25y² = H + K

=> H = 9x² - 25y² - K

putting (5,0) in H

=> 9(5)²-25(0) - K = 0

=> K = 225

Hence, H : 9x² - 25y² = 225 is the equation of hyperbola.

Now, (5,0) and (-5,0) passes through H.