Step by step solution to "Find the two points of intersection of the..."

Laura Jenkins

Answered question

2022-03-01

Find the two points of intersection of the circles:

x^{2} + y^{2} + 5 x + y - 26 = 0

x^{2} + y^{2} + 2 x - y - 15 = 0

besplodnexkj

Beginner2022-03-02Added 7 answers

Step 1
Multiply all terms in the first equation by -1 to obtain an equivalent equation and keeping the second equation unchanged.

- x^{2} - y^{2} - 5 x - y + 26 = 0

and

x^{2} + y^{2} + 2 x - y - 15 = 0

Adding the same sides of the two equations to obtain a linear equation,

- 5 x - y + 26 + 2 x - y - 15 = 0

Simplifying,

- 3 x - 2 y + 11 = 0

3 x + 2 y - 11 = 0

Solving for x,

3 x = 11 - 2 y

x = \frac{11 - 2 y}{3}

Step 2
Substituting the value of x in equation (1),

{(\frac{11 - 2 y}{3})}^{2} + y^{2} + 5 (\frac{11 - 2 y}{3}) + y - 26 = 0

Simplifying,

\frac{{(11 - 2 y)}^{2}}{9} + y^{2} + \frac{5 (11 - 2 y)}{3} + y - 234 = 0

{(11 - 2 y)}^{2} + 9 y^{2} + 15 (11 - 2 y) + 9 y - 234 = 0

121 - 44 y + 4 y^{2} + 9 y^{2} + 165 - 30 y + 9 y - 234 = 0

13 y^{2} - 65 y + 286 - 234 = 0

13 y^{2} - 65 y - 52 = 0

y^{2} - 5 y + 4 = 0

Solving quadratic equation,

(y - 1) (y - 4) = 0

\Rightarrow y = 1, 4

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