Question

Find the point on the line\(y = 2x + 3\)that is closest to the origin.

Short answer

The point closest to the origin is \(\left( { - \frac{6}{5},\frac{3}{5}} \right)\)

Step-by-step solution

 Step 1: Given Data

Equation of the line is \(y = 2x + 3\)

 Step 2: Determination of the point closest to the origin

The closest point will be perpendicular distance from origin to the line.

Therefore, find equation of the perpendicular from \(\left( {0,0} \right)\)on line.

\(y = 2x + 3\)……(1)

The equation of line is:

\(y = mx + C\)

Slope, \(m = 2\).

Slope of the perpendicular \( = - \frac{1}{m} = - \frac{1}{2}\)

The perpendicular equation is found by

\(\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)\)

\(\left( {y - 0} \right) = - \frac{1}{2}\left( {x - 0} \right)\)

\(y = - \frac{1}{2}x\)

\(2y + x = 0\)…………(2)

Solving equation (1) and (2) as:

\(5y = 3\)

\(y = \frac{3}{5}\)

Find the value of x as:

\(\begin{aligned}{c}2y + x &= 0\\x + 2\left( {\frac{3}{5}} \right) &= 0\\x &= - \frac{6}{5}\end{aligned}\)

\(x = - \frac{6}{5}\)and \(y = \frac{3}{5}\)

Thus, the point on the line is \(\left( { - \frac{6}{5},\frac{3}{5}} \right)\).