What is the equation of a line, in standard form, perpendicular to the line [tex] y=\frac{4}{3} x-16 [/tex] that goes through the point [tex] (1,-9) [/tex]?

A. [tex] 4x - 3y = 33 [/tex]
B. [tex] 3x + 4y = -33 [/tex]
C. [tex] 4x + 3y = -33 [/tex]
D. None of these are correct.



Answer :

To determine the equation of a line perpendicular to the given line [tex]\(y = \frac{4}{3} x - 16\)[/tex] and passing through the point [tex]\((1, -9)\)[/tex], follow these steps:

1. Find the slope of the given line:

The given equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope. So, the slope of the given line is:
[tex]\[ \text{slope} = \frac{4}{3} \][/tex]

2. Determine the slope of the perpendicular line:

The slope of a line perpendicular to another is the negative reciprocal of the original slope:
[tex]\[ \text{slope of perpendicular line} = -\frac{1}{\left(\frac{4}{3}\right)} = -\frac{3}{4} \][/tex]

3. Use the point-slope form to find the equation of the perpendicular line:

The point-slope form of a line's equation is:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the given point [tex]\((1, -9)\)[/tex] and [tex]\(m\)[/tex] is the slope of the perpendicular line [tex]\(-\frac{3}{4}\)[/tex]. Plugging in these values:
[tex]\[ y - (-9) = -\frac{3}{4}(x - 1) \][/tex]
Simplifying:
[tex]\[ y + 9 = -\frac{3}{4}x + \frac{3}{4} \][/tex]

4. Convert the equation to slope-intercept form:

Subtract 9 from both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{4} - 9 \][/tex]
Simplify the constant term on the right-hand side:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{4} - \frac{36}{4} \][/tex]
[tex]\[ y = -\frac{3}{4}x - \frac{33}{4} \][/tex]

5. Convert the equation to standard form [tex]\(Ax + By = C\)[/tex]:

To eliminate the fractions, multiply every term by 4:
[tex]\[ 4y = -3x - 33 \][/tex]
Rearrange to bring all terms to one side:
[tex]\[ 3x + 4y = -33 \][/tex]

Thus, the correct equation of the line in standard form, perpendicular to the line [tex]\(y = \frac{4}{3}x - 16\)[/tex] and passing through the point [tex]\((1, -9)\)[/tex], is:
[tex]\[ 3x + 4y = -33 \][/tex]