Prove that the curve y^2=x^2-9 and the line y=x-1 are intersecting.

At the points where two curves intersect the x and y coordinates
are the same.

Now we have y^2 = x^2 - 9 and y = x -
1.

y^2 = x^2 - 9

=> (x - 1)^2 =
x^2 - 9

=> x^2 + 1 - 2x = x^2 -
9

=> -2x = -10

=> x =
5

y = x - 1 = 4

We see that x + 1 only
touches the curve y^2 = x^2 - 9 at one point (5, 4).

y
= x - 1 is a tangent to y^2 = x^ - 9 at the point (5,4).