Verify if the curve y2x290 and the line yx10 have common poi

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

Solution

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).