How to Solve this Question?

Question

1. Find the coordinates of the points of intersection of the line 2x + 3y + 2 = 0 with the curve 2xy = - 1

2. The line 2x + 3y = 1 intersects the curve x(x + y) = 10 at A and B. Calculate the coordinates of A and of B

Answer 1

2x + 3y + 2 = 0

2xy = -1

2x = -1/y

-1/y + 3y +2 = 0

-1 + 3y^2 +2y = 0

3y^2 + 2y - 1 = 0

(3y -1)(y +1) = 0

y = -1, 1/3

2x = -1/y

x = -1/2y = -1/2(-1) or -1/2(1/3) = 1/2 or -3/2

intersections are (1/2, -1) and (-3/2, 1/3)

or (0.5, -1) and (-1.5, 0.333...)

2x+3y = 1

x(x+y) = 10

3y = 1-2x

y = (1-2x)/3 = 1/3 -2x/3

x(x+(1-2x)/3) = 10

x(x + 1/3 - 2x/3) = 10

x^2 +x/3 - 2x^2/3 = 10

3x^2 +x -2x^2 = 30

x^2 +x -30 = 0

(x+6)(x-5) = 0

x =5 or -6

y = 1/3 -2x/3 = 1/3 -10/3 or 1/3 +12/3 = -3 or 13/3

solutions are (5, -3) and (-6, 13/3)

A = (5,-3), B = (-6, 13/3)