At what points does the line y = x +1 intersect the circle x^2 + y^2 + 18x + 20y + 81 = 0.

The method we shall use is substitution. We shall replace y each term by 'x + 1' so we are able to obtain an equation in one variable. x^2 + (x+1)^2 + 18x + 20(x+1) + 81 = 0. We now expand the brackets. x^2 + x^2 + 2x + 1 + 18x + 20x + 20 +81 = 0. Now we shall collection like terms to arrive at 2x^2 + 40x + 102 = 0. We notice that every term is a multiple of 2 so we can divide through by 2 to give x^2 + 20x + 51 = 0. We can use the quadratic formula to solve this or we can spot it factorises to (x + 17)(x + 3) = 0. A trick to spotting this is that the two numbers in the brackets must be factors of 51 and give 20 when added together. Thus we know the x-coordinates of the points are x = -3 and x = -17, as these are the x values which make the expression equal 0. To find the y-coordinates we can use the fact y = x+1, thus the y-coordinates are y = -3 + 1 = -2 and y = -17 + 1 = -16. So the coordinates of intersection are (-3, -2) and (-17, -16).