Mathematics

Problem

Two pins are fixed at the points

F_{1}(-13,0)

and

F_{2}(13,0)

on a smooth board. A string

40\,\text{cm}

long is stretched tight around the pins and a pencil

P

moves with the string always taut. a. (a) Prove that the locus of

P

is an ellipse, and determine its equation in standard form. b. (b) Calculate the lengths of its major and minor axes. c. (c) Find the two possible

y

-coordinates of

P

when its

x

-coordinate is

9\,\text{cm}