Find the points of intersection of the line y=2x + 1

Question

 and the circle (x-7)^2 + y^2 = 125? 

 

 

Answer 1

The line equation y = 2x + 1 ---> (1)

The circle equation (x - 7)² + y² = 125

Solve the circle equation for y.

y² = 125 - (x - 7)²

Apply square root on each side.

y = ± √[125 - (x - 7)²] ---> (2)

 

Equate the y form equations (1) and (2), then solve for x.

2x + 1 = ± √[125 - (x - 7)²]

squaring on both sides.

(2x + 1)² = { ± √[125 - (x - 7)²]}²

4x² + 1 + 4x = 125 - (x - 7)²

4x² + 1 + 4x - 125 + (x - 7)² = 0

4x² + 1 + 4x - 125 + x² + 49 - 14x = 0

5x² - 10x - 75 = 0

x² - 2x - 15 = 0

x² - 5x + 3x - 15 = 0

x(x - 5) + 3(x - 5) = 0

(x - 5)(x + 3) = 0

x = 5 and x = - 3

 

Substitute the above x values in either of two equations.

Substitute x = 5 in y = 2x + 1

y = 2(5) + 1

y = 11

Substitute x = - 3 in y = 2x + 1

y = 2(- 3) + 1

y = - 5

The intersection points are (x,y) = (5, 11) and (x,y) = (- 3, - 5).