The line (2y = x + 3) meets the circle (x^{2} + y^{2} – 2x + 6y – 15 = 0) at points M and N, where N is in the first quadrant. Find the coordinates of M and N.
Explanation
Line : (2y = x + 3)
(therefore y = frac{1}{2} (x + 3))
(x^{2} + y^{2} – 2x + 6y – 15 = 0)
(x^{2} + (frac{1}{2} (x + 3))^{2} – 2x + 6(frac{1}{2} (x + 3)) – 15 = 0)
(x^{2} + frac{x^{2}}{4} + frac{6x}{4} + frac{9}{4} + 3x + 9 – 15 = 0)
(frac{5x^{2}}{4} + frac{10x}{4} – frac{15}{4} = 0)
(equiv 5x^{2} + 10x – 15 = 0)
(5x^{2} – 5x + 15x – 15 = 0 implies 5x(x – 1) + 15(x – 1) = 0)
((5x + 15)(x – 1) = 0 implies text{x = -3 or 1})
When (x = -3), (y = frac{1}{2} (-3 + 3) = 0)
When (x = 1), (y = frac{1}{2} (1 + 3) = 2)
The coordinates are M(-3, 0) and N(1, 2).