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

justaguide | Certified Educator

The curve y^2 - x^2 + 9 = 0 and the line y - x + 1 = 0 have common points if the two equations give a solution.

y - x + 1 = 0

=> x = y + 1

Substitute in y^2 - x^2 + 9 = 0

=> y^2 - y^2 - 1 - 2y + 9 = 0

=> -2y + 8 = 0

=> y = 4

x = 5

The common point of the two is (5 , 4)

giorgiana1976 | Student

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.

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).