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In order to calculate this integral we must use the sustitution provided. x=2siny. Firtsly I will differentiate to find the dx component of the integral, so dx/dy=2cosy hence, dx=2cosydy. Now for the limi...

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