What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point (2, 5)?

A. [tex]\( y + 5 = x + 2 \)[/tex]
B. [tex]\( y - 2 = x - 5 \)[/tex]
C. [tex]\( y - 5 = -(x - 2) \)[/tex]
D. [tex]\( y + 2 = -(x + 5) \)[/tex]



Answer :

To find the equation of the line that is perpendicular to the given line and passes through the point [tex]\((2, 5)\)[/tex], follow these steps:

1. Determine the slope of the given line:
The given line is in the form [tex]\(y - y_1 = m(x - x_1)\)[/tex]. Rewriting the original line [tex]\(y + 2 = x + 5\)[/tex] into the point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex]:
[tex]\[ y - (-2) = 1(x - (-5)) \][/tex]
So, the slope [tex]\(m\)[/tex] of the given line is 1.

2. Find the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. The negative reciprocal of 1 is -1.

3. Use the point-slope form to write the equation:
The point-slope form of a line's equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Given the point [tex]\((2, 5)\)[/tex] and the slope [tex]\(-1\)[/tex], substitute these values into the point-slope form:
[tex]\[ y - 5 = -1(x - 2) \][/tex]

Therefore, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((2, 5)\)[/tex] in point-slope form is:
[tex]\[ y - 5 = -1(x - 2) \][/tex]

So, the correct choice from the options given is:
[tex]\[ y - 5 = -(x - 2) \][/tex]