Mathematics

Problem

For the parabola

y^{2}=4ax

with

a>0

, a point

P(k,6)

is known to satisfy two conditions: 1.

P

lies on the parabola. 2. The distance

PF

from

P

to the focus

F(a,0)

10

units. a. (i) Using the fact that

P

lies on the parabola, establish a relation between

k

and

a

. b. (ii) Write the distance formula for

PF

and, using the given numerical value of

PF

, obtain a second relation between

k

and

a

. c. (iii) Solve the simultaneous relations from parts (i) and (ii) to determine the admissible positive values of

a

and the corresponding coordinates of

P

. d. (iv) Verify for each solution that

PF

equals the perpendicular distance from

P

to the directrix, thereby completing the check that

P

indeed lies on the parabola.