Find the point on the line y = 4x + 9 that is closest to the origin. (Give your answers correct to three decimal places.)

Question

Find the point on the line y = 4x + 9 that is closest to the origin. (Give your answers correct to three decimal places.)

Answer 1

The distance between (0,0) and ( x, 4x+9) is

by distance formula,

sqrt ( (x-0)^2 + (4x+9-0)^2)

= sqrt ( x^2 + (4x+9)^2)

= sqrt( x^2 + 16x^2 + 72x + 81)

= sqrt( 17x^2 + 72x + 81)

= (17x^2 + 72x + 81)^(1/2)

= D(x)

is distance between any point on the line and the origin

as a function of x

Minimizing:

0 = dD/dx = (1/2)(17x^2 + 72x + 81)^(-1/2) ( 34x + 72)

= (17x + 36)/sqrt( 17x^2 + 72x + 81)

the solution is 0 = 17x + 36

= -36/17