Simultaneous Equations When A Straight Line Intersects A Circle Take the following simultaneous equations: x2 + y2 = 25! ! ! Eq 1 x + y = 1! ! ! Eq 2 y = 1 - x! ! ! Eq 2 ! Eq 1 Rearrange Eq 2 in terms of x: Substitute Eq 2 into Eq 1: x2 + (1 - x)2 = 25! Expand out the brackets in Eq 1 and simplify to give a quadratic equation: x2 + (1 - x)(1 - x) = 25! Eq 1 x2 + 1 - 2x + x2 = 25!! Eq 1 2x2 - 2x + 1 = 25! ! Eq 1 2x2 - 2x - 24 = 0! ! Eq 1 Solve Eq 1 using the quadratic formula to find the x values of the points of intersection: ax2 + bx + c = 0 a=2 b = -2 c = -24 x = -b ±√(b2 - 4ac) 2a x = -(-2) ±√((-2)2 - (4 X 2 X -24)) 2X2 x = 2 ±√(4 - (-192)) 4 x = 2 ±√(196) 4 x = 2 ± 14 4 x = 4 or -3 Substitute the x values into Eq 2 to find the y values that correspond with the x value solutions: If x = 4, y = 1 - 4 = -3 If x = -3, y = 1 - (-3) = 4 Solutions are ! and ! ! ! x = 4, y = -3 x = -3, y = 4 www.greatmathsteachingideas.com © 2010 All Rights Reserved