Question

Find the distance from the point \((5,3)\) to the line with equation \(2 x-y+3=0\) Hint: Find the point of intersection of the given line and the line perpendicular to it that passes through \((5,3)\).

Short answer

The distance from the point \((5,3)\) to the line \(2x-y+3=0\) is \(\frac{\sqrt{545}}{5}\).

Step-by-step solution

Determine the slope of the given line

To find the slope of the given line, we need to rewrite the linear equation \(2x-y+3=0\) in the slope-intercept form, i.e., \(y=mx+b\). In this form, \(m\) signifies the slope of the line. Rearrange the equation \[y = 2x + 3\] Now we can see that the slope of the given line is \(m = 2\).

Determine the slope of the line perpendicular to the given line

We know that if two lines are perpendicular, the product of their slopes is equal to \(-1\). Therefore, let the slope of the line perpendicular to the given line be \(m_p\). We have: \[m \cdot m_p = -1\] Substituting the slope of the given line (m = 2) in the formula, we get: \[2 \cdot m_p = -1\] \[m_p = -\frac{1}{2}\]

Find the equation of the line perpendicular to the given line

We have the slope of the perpendicular line \(m_p = -\frac{1}{2}\), and we know that it passes through the point \((5,3)\). Using the point-slope form of a linear equation, \(y-y_1 = m_p(x-x_1)\), we can find the equation of the perpendicular line. Here, \((x_1,y_1) = (5,3)\). Thus, we get \[y - 3 = -\frac{1}{2} (x - 5)\]

Find the point of intersection of the two lines

To find the intersection point \((x_i, y_i)\) of the given line and the perpendicular line, we need to solve the linear equations simultaneously. The given line: \[y = 2x + 3\] The perpendicular line: \[y - 3 = -\frac{1}{2} (x - 5)\] We can solve these equations using substitution. Substitute the perpendicular line equation into the given line equation: \[2x + 3 = 3 -\frac{1}{2} (x - 5)\] Solve for \(x_i\): \[2x +3 = 3 - \frac{1}{2}x + \frac{5}{2}\] \[\frac{5}{2}x = \frac{2}{2}\] \[x_i = \frac{2}{5}\] Now substitute the value of \(x_i\) back into either of the equations. We can use the given line: \[y_i = 2\left(\frac{2}{5}\right) + 3\] \[y_i = \frac{4}{5} + \frac{15}{5}\] \[y_i = \frac{19}{5}\] So, the point of intersection is \(\left(\frac{2}{5}, \frac{19}{5}\right)\).

Calculate the distance between the given point and the point of intersection

Lastly, we use the distance formula to find the distance between the given point \((5,3)\) and the point of intersection \(\left(\frac{2}{5}, \frac{19}{5}\right)\): \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\] Substitute the points into the distance formula: \[d = \sqrt{\left(\frac{2}{5} - 5\right)^2 + \left(\frac{19}{5} - 3\right)^2}\] \[d = \sqrt{\left(\frac{-23}{5}\right)^2 + \left(\frac{4}{5}\right)^2}\] \[d = \sqrt{\frac{529 + 16}{25}}\] \[d = \sqrt{\frac{545}{25}}\] \[d = \frac{\sqrt{545}}{5}\] Thus, the distance from the point \((5,3)\) to the line \(2x-y+3=0\) is \(\frac{\sqrt{545}}{5}\).