The document discusses finding the gradient of the tangent and normal lines to the curve y^2=4ax at a point P(p, y). It is shown that the gradient of the tangent is -y/2a and the gradient of the normal is -2a/y. The locus of the midpoints R of the normal PQ as P varies is derived to be x^2+4ay=4a^2.

1. Further Coordinate Geometry WJEC FP2 June 2008

3. Find the gradient of the tangent by differentiating implicitly Differentiating both sides with respect to x y We will use the general equation of a line At the point P x

4. So the GRADIENT OF THE TANGENT AT POINT P IS The GRADIENT OF THE NORMAL IS THEREFORE Because the product of the gradients of TANGENT and NORMAL is -1

5. This is the required EQUATION OF THE NORMAL QED We will use the general equation of a line

6. The normal meets the x axis at Q when y=0 find the x coordinate on the normal Q is the point

7. R is the midpoint of PQ FIND THE MIDPOINT X Coordinate of R Y Coordinate of R (THE MIDPOINT) R is

8. To find the LOCUS OF R as p varies we ELIMINATE p and SUBSTITUTE This is the equation of the LOCUS of R as p varies.