Write an equation of a line that passes through [tex]\((1,4,-3)\)[/tex] and is perpendicular to the line that passes through [tex]\((-4,7)\)[/tex] and [tex]\((9, 8)\)[/tex].

A. [tex]\(y=13 x-15.2\)[/tex]

B. [tex]\(y=\frac{1}{13} x+15.2\)[/tex]

C. [tex]\(y=-\frac{1}{13} x-15.2\)[/tex]

D. [tex]\(y=-13 x+15.2\)[/tex]



Answer :

To write the equation of a line that passes through (1, 4) and is perpendicular to the line that passes through the points (-4, 7) and (9, 8), we need to follow these steps:

1. Find the slope of the given line:

The points (-4, 7) and (9, 8) describe a line. The slope [tex]\(m\)[/tex] of this line is found using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, [tex]\((x_1, y_1) = (-4, 7)\)[/tex] and [tex]\((x_2, y_2) = (9, 8)\)[/tex].

Calculating this, we get:
[tex]\[ m = \frac{8 - 7}{9 - (-4)} = \frac{1}{13} \][/tex]

2. Determine the slope of the perpendicular line:

The slope of a line perpendicular to the one calculated is the negative reciprocal of the original slope. So if the original slope is [tex]\(\frac{1}{13}\)[/tex], the slope of the perpendicular line will be:
[tex]\[ m_{\text{perp}} = -13 \][/tex]

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

We know this line passes through the point (1, 4). The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Plugging in [tex]\((x_1, y_1) = (1, 4)\)[/tex] and [tex]\(m = -13\)[/tex]:
[tex]\[ y - 4 = -13(x - 1) \][/tex]

4. Simplify the equation:

Distribute and simplify to get the equation in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y - 4 = -13x + 13 \implies y = -13x + 13 + 4 \implies y = -13x + 17 \][/tex]

From the given choices, none of them exactly match [tex]\(y = -13x + 17\)[/tex]. It seems we have a discrepancy, there might be a miscommunication in choice matching. Based on the correct perpendicular slope, the best fit based on our derived slope would be:

[tex]\[ y = -13x + 15.2 \][/tex]
The correct answer closest to the reasoning would be:
[tex]\[ y = -13x + 15.2 \][/tex]