2014 PAPER 2 – QUESTION 3.

This question combines the concepts of the line and the circle.

(a)(i) The circle c has equation

(x+2)2 + (y-3)2 = 100. Write down the coordinates of A, the centre of c. Write down r, the length of the radius of c. For the centre, switch the sign of the numbers in the brackets. For the radius, find the square root of the value on the right-hand side. Leave this as a square root when possible. Do not change the number to decimal form. (ii) Show that the point P(–8 ,11) is on the circle c.

Substitute the point (-8, 11) into the equation of a circle:

2 2

(–8+2) + (11–3) =100

36+64=100

100 = 100 i.e. the left-hand side is equal to the right-hand side. Therefore, the point (–8, 11) is on the circle (x+2)2 + (y-3)2 = 100. It is very important to include this concluding statement as marks can be easily lost. (b)(i) Find the slope of the radius [AP]. (ii) Hence, find the equation of t, the tangent tocatP.

Recall that tangents are perpendicular to the radius that joins the centre of a circle to the point of tangency. This fact is used to find the slope of the tangent. Change the sign of the slope and flip the fraction upside down. Thus the perpendicular slope is m = Sub this slope and the point P(–8, 11) into the equation of a line formula: