Verify if the curve y^2-x^2+9=0 and the line y-x+1=0 have common points?

To verify if the curve and the line are intercepting each
other, w'ell have to solve the system of equations of the curve and the
line.

We'll start with the equation of the curve and we'll
add 9 both sides:

x^2 - y^2 =
9

We'll recognize the difference of
squares:

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

(x
- y)(x + y) = 9

We'll re-write the second
equation:

x - y = 1 (2)

We'll
replace the value of the second equation into the
first:

1*(x+y)=9

x + y = 9
(3)

We'll compute (2)+(3) to eliminate
y:

x - y + x + y = 1 + 9

2x =
10

x = 5

We'll substitute x =
5 into (2):

x - y = 1 <=> 5 - y = 1 =>
y = 5 - 1 => y = 4

The curve and the
line are intercepting each other at the point of coordinates (5 ,
4).