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).
To prove that the line and the curve are intercepting, we'll have to check if the system formd by the equations of the line and the curve, has any solutions.
We'll write the first equation as a difference of two squares:
We'll substitute the second equation into the first:
x + y = 9
We'll change the second equation and we'll write y with respect to x.
y = x - 1
But x + y = 9
x + x - 1 = 9
2x - 1 = 9
2x = 10
x = 5
y = 5 - 1
y = 4