Problem 21 - The Calculus with Analytic Geometry

Question

Given the line \(l\) having the equation \(2 y-3 x=4\) and the point \(P(1,-3)\), find (a) an equation of the line through \(P\) and perpendicular to \(l_{;}^{l}\) (b) the shortest distance from \(P\) to line \(l\).

Step-by-Step Solution

Answer

a) The equation of the line is \(y = -\frac{2}{3}x - \frac{7}{3}\). b) The shortest distance is \(\sqrt{13}\).

Step 1: Find the slope of the given line

For the line with equation \(2y - 3x = 4\), rewrite it in slope-intercept form \(y = mx + c\). First, solve for \(y\): \[ 2y = 3x + 4 \] \[ y = \frac{3}{2}x + 2 \] Therefore, the slope of the line \(l\) is \(\frac{3}{2}\).

Step 2: Determine the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the other line's slope. The slope of line \(l\) is \(\frac{3}{2}\), so the slope of the perpendicular line will be \(-\frac{2}{3}\).

Step 3: Find the equation of the line through point \(P\)

Using the point-slope form of the line \((y - y_1 = m(x - x_1))\), where \((x_1, y_1) = (1, -3)\) and \(m = -\frac{2}{3}\), the equation of the line becomes: \[ y - (-3) = -\frac{2}{3}(x - 1) \] Simplify this to get: \[ y + 3 = -\frac{2}{3}x + \frac{2}{3} \] \[ y = -\frac{2}{3}x - \frac{7}{3} \] So, the equation of the line through point \(P\) and perpendicular to \(l\) is \(y = -\frac{2}{3}x - \frac{7}{3}\).

Step 4: Use the distance formula

The shortest distance from a point to a line can be found using the formula: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] For the line \(2y - 3x = 4\), rewrite it in standard form where \(A = -3, B = 2, \text{ and } C = -4\). Using point \((1, -3)\), substitute into the distance formula: \[ d = \frac{|-3(1) + 2(-3) - 4|}{\sqrt{(-3)^2 + 2^2}} \] \[ d = \frac{|-3 - 6 - 4|}{\sqrt{9 + 4}} \] \[ d = \frac{|-13|}{\sqrt{13}} \] \[ d = \frac{13}{\sqrt{13}} \] \[ d = \sqrt{13} \] So, the shortest distance from point \(P\) to line \(l\) is \(\sqrt{13}\).

Key Concepts

slope-intercept form

The slope-intercept form is incredibly useful for equations of straight lines. This form is given as: y = mx + c, where:

• m is the slope of the line
• c is the y-intercept (the value of y when x=0)

To convert a linear equation to slope-intercept form, isolate y on one side. For example, our given line equation is 2y - 3x = 4. By solving for y, we rewrite it as: 2y = 3x + 4
y = \(\frac{3}{2}\)x + 2.
Edit the given text appropriately for URL and 'id' slug. Here, the slope m is \(\frac{3}{2}\), and the y-intercept c is 2.

distance formula

The distance formula calculates the shortest distance between a point and a line in a 2-dimensional plane. The distance formula for a point \( (x_1, y_1) \) to a line Ax + By + C = 0 is given by: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] In our example, the line's equation 2y - 3x = 4 is first rewritten in standard form as -3x + 2y - 4 = 0, where:

• A = -3
• B = 2
• C = -4

Inserting the point P(1, -3) into the formula gives us the distance as \( d = \sqrt{13} \): \[ d = \frac{|-3(1) + 2(-3) - 4|}{\sqrt{(-3)^2 + 2^2}} \] \[ d = \frac{|-3 - 6 - 4|}{\sqrt{9 + 4}} \] \[ d = \frac{13}{\sqrt{13}} \]\[ d = \sqrt{13} \]

negative reciprocal

When working with perpendicular lines, the slopes are related by the concept of the negative reciprocal. If one line has a slope m, the slope of the perpendicular line is \(-\frac{1}{m}\). For instance, if the slope of our given line l is \(\frac{3}{2}\), then the slope of the line perpendicular to it is: \[ -\frac{1}{\left(\frac{3}{2}\right)} = -\frac{2}{3} \].

point-slope form

Point-slope form is another way to write the equation of a line, especially useful when you know one point on the line and its slope. The form is given by: \( y - y_1 = m(x - x_1) \), where:

• \( (x_1, y_1) \) is a point on the line
• m is the slope

For the point P(1, -3) and the slope -\(\frac{2}{3}\), the equation becomes: \( y - (-3) = -\frac{2}{3}(x - 1) \) Simplifying, we get: \( y + 3 = -\frac{2}{3}x + \frac{2}{3} \) = \( y = -\frac{2}{3}x - \frac{7}{3} \). So, the line through P and perpendicular to line l has the equation: \( y = -\frac{2}{3}x - \frac{7}{3} \).