Mathematics

Problem

Let

P(8,0)

be a fixed point in the plane. For each real number

t

, let

Q_t

be the point that lies on the line

y=5-2x

with

x

-coordinate

t

. Your task is to locate the point

Q_t

that is nearest to

P

. (a) Express the square of the distance

d^2(t)

between

P

and

Q_t

as a function of

t

. Show all algebraic steps. (b) Determine the value of

t

that minimizes

d^2(t)

. Justify that your result gives a minimum rather than a maximum. (c) State the coordinates of the closest point

Q

and compute the minimal distance

d_{\min}

. Give exact values.