Question

Prove that the line y = x + 2 touches the hyperbola 5x2 - 9y2 = 45 and also find the point of contact.

Answer 1

The given hyperbola is x2/a2 - y2/b2 = 1

For the line y = x + 2, we have c = 2 and m = 1. Now,

a2m2 − b2 = 9(1) − 5 = 4 = c2

Hence, the line y = x + 2 touches the hyperbola. Suppose (x1, y1) is the point of contact. So

xx1/9 - yy1/5 = 1

This line and y = x + 2 represent the same line. Therefore

x1/9/1 = -y1/5/-1 = -1/2

Hence

x1 = -9/2 and y1 = -5/2

The point of contact is (-9/2, -5/2).