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

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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).

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