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]
