The question I am stuck on is (iii):

This part concerns the conic with equation

(i) By rearranging the equation of the conic, classify it as an ellipse, parabola or hyperbola in standard position, and sketch the curve.

I have re-arranged the equation and discovered it is an ellipse in standard position:

a=5, b=

The points of the ellipse are (-5,0), (0, ), (5,0) and (0, )

(ii) Find exact values for the eccentricity, foci and directrices of this conic, and mark the foci and directrices on your sketch.

eccentricity (e) = =

foci = +-ae = and

directrices = +-a/e =

(iii) Check your answers to part (a)(ii) by verifying that the equation PF = ePd holds at each of the two points P where the conic intersects the x-axis, where F is the focus with negative x-coordinate, d is the corresponding directrix and e is the eccentricity.

point, P where the conic intersect the x-axis ... etc. (-5,0)

focus with negative x co-ordinate =

corresponding directrix =

eccentricity =

I know that PF = a-ae =

and ePd = e(a-ae)/e =

=

so PF = ePd, but I'm not sure what they mean when they say the two points, P where the conic intersects the x-axis ... etc.

I hope what I have done so far is correct? Thanks again for your help. It is most appreciated (Happy)