Two radio transmitters are located at and on a Cartesian plane (distances in kilometres)

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Two radio transmitters are located at

T_1(-6,0)

and

T_2(6,0)

on a Cartesian plane (distances in kilometres). A point

P

on the plane receives the signals such that the difference of its distances from

T_1

and

T_2

is always

6

km. (i) Show that the locus of

P

is a hyperbola. Determine the values of

a

and

c

for this hyperbola. (ii) Find the length of the semi-conjugate axis

b

. (iii) Write the equation of the hyperbola. (iv) Compute the eccentricity of the hyperbola and interpret its value in the context of this situation.