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To find potential points of intersection between the line and the circle, we need to solve the equations simultaneously. So, we substitute y = x - 7 into the equation of the circle: (x + 2)² + ...

Because the value, x and its inverse, -1/x should multiply together to give -1. This is proof that two lines are perpendicular (or having the same magnitude of gradient but opposite signs +/- ).

Grouping is essentially factorisation, it's just the idea that ax + bx = (a+b)x, which we already know. But the difficulty is in recognising this relationship when x is more complicated, for example when ...

Multiply out the brackets:

3a+12=ac+5f

3a-ac=5f-12

a(3-c)=5f-12

a=5f-12/3-c

The first step is to split the fraction into 2 separate fractions using partial fractions techniques. Write 3x+3/2x^2+3x as A/x + B/2x+3 and solve to get A = 1, B = 1. We have now converted 3x+3/2x^2+3x i...