OCR Advanced FSMQ - Additional Maths

6.12 A quicker way to determine the equation of a line through a point that is parallel to another line Suppose you want to find the equation of the straight line passing through the point (1, 4) which is parallel to the line having equation 4 x + 2 y = 5. A line parallel to the equation 4 x + 2 y = 5 will have an equation of the form 4 x + 2 y = c . You now substitute the coordinates of the point through which the line passes, into this equation. Hence, we have 4 x + 2 y = c 4(1) + 2(4) = c , giving c = 12. This value is substituted back into the equation, so the equation of the line is 4 x + 2 y = 12 Examples 11 Find the equation of the line parallel to the line 2 x − 5 y + 4 = 0, if the line passes through the point (3, 1). Answer 11 Let the equation of the parallel line be 2 x − 5 y + c = 0. Substituting the coordinates (3, 1) into this equation gives 2(3) − 5(1) + c = 0 1 + c = 0 Hence, c = −1. The equation of the required line is 2 x − 5 y − 1 = 0 12 P is the point (0, 6) and Q is the point (5, p ). (a) (i) Find the gradient of the line with equation 2 x + 5 y = 40. (ii) Find the equation of the line through P which is parallel to the line 2 x + 5 y = 40. (b) The line through P also passes through the point (5, p ). Find the value of p . Notice the way the number on its own is replaced with c . Here x = 1 and y = 4 are substituted into the equation. The number on its own is replaced by c . 6 Coordinate geometry of straight lines 133