Select the correct answer.

Which equation represents the line that is perpendicular to [tex] y=\frac{4}{5} x+23 [/tex] and passes through [tex] (-40,20) [/tex]?

A. [tex] y=-\frac{5}{4} x-15 [/tex]
B. [tex] y=-\frac{5}{4} x-30 [/tex]
C. [tex] y=\frac{4}{5} x+52 [/tex]
D. [tex] y=\frac{4}{5} x-56 [/tex]



Answer :

To determine the equation of a line that is perpendicular to the line [tex]\( y = \frac{4}{5} x + 23 \)[/tex] and passes through the point [tex]\((-40, 20)\)[/tex], follow these steps:

1. Identify the slope of the given line:

The equation of the given line is [tex]\( y = \frac{4}{5} x + 23 \)[/tex]. The slope ([tex]\( m_1 \)[/tex]) of this line is [tex]\(\frac{4}{5}\)[/tex].

2. Determine the slope of the perpendicular line:

Lines that are perpendicular have slopes that are negative reciprocals of each other. This means if one line has a slope of [tex]\( m \)[/tex], the perpendicular line will have a slope of [tex]\(-\frac{1}{m} \)[/tex].

Therefore, the slope ([tex]\( m_2 \)[/tex]) of the line perpendicular to our given line will be:
[tex]\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{4}{5}} = -\frac{5}{4} \][/tex]

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

The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.

Substituting the point [tex]\((-40, 20)\)[/tex] and the slope [tex]\(-\frac{5}{4}\)[/tex]:

[tex]\[ y - 20 = -\frac{5}{4}(x + 40) \][/tex]

4. Simplify the equation:

Distribute the slope on the right-hand side:
[tex]\[ y - 20 = -\frac{5}{4}x - \frac{5}{4} \cdot 40 \][/tex]

Simplify inside the parentheses:
[tex]\[ y - 20 = -\frac{5}{4}x - 50 \][/tex]

Finally, solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{5}{4}x - 50 + 20 \][/tex]
[tex]\[ y = -\frac{5}{4}x - 30 \][/tex]

Thus, the equation of the line that is perpendicular to the given line and passes through [tex]\((-40, 20)\)[/tex] is:
[tex]\[ y = -\frac{5}{4}x - 30 \][/tex]

Therefore, the correct answer is:
B. [tex]\( y = -\frac{5}{4}x - 30 \)[/tex]