The orthogonal projection of the point (2 , - 3) on the line x + y = 0 isa)(2 , - 3)b)(2 , 3)c)(-2 , -3)d)(5/2 , - 5/2)Correct answer is option 'D'. Can you explain this answer?

Question

Accepted Answer

Answer

D

Answer

Orthogonal Projection:
The orthogonal projection of a point onto a line is the closest point on the line to the given point, measured along a line perpendicular to the given line.

Given:
Point P(2, -3)
Line: x + y = 0

To find the orthogonal projection of point P onto the line x + y = 0, we need to find the point on the line that is closest to point P.

Step 1: Find the slope of the line
The given line can be written in slope-intercept form as:
y = -x
Comparing this equation with y = mx + c, we can see that the slope (m) of the line is -1.

Step 2: Find the equation of the line perpendicular to the given line
Since the line perpendicular to the given line has a slope that is the negative reciprocal of the given line's slope, the slope of the perpendicular line is 1.

Step 3: Find the equation of the perpendicular line passing through point P
Using the point-slope form of a line, we can write the equation of the perpendicular line passing through point P(2, -3) as:
y - (-3) = 1(x - 2)
y + 3 = x - 2
y = x - 5

Step 4: Find the intersection point of the given line and the perpendicular line
To find the intersection point, we need to solve the system of equations formed by the given line x + y = 0 and the perpendicular line y = x - 5.

Substituting y from the second equation into the first equation, we get:
x + (x - 5) = 0
2x = 5
x = 5/2

Substituting x back into the second equation, we get:
y = (5/2) - 5
y = -5/2

Therefore, the intersection point of the given line x + y = 0 and the perpendicular line y = x - 5 is (5/2, -5/2).

Step 5: Verify if the intersection point is the orthogonal projection of point P
To verify if the intersection point (5/2, -5/2) is the orthogonal projection of point P(2, -3), we can calculate the distance between these two points and compare it with the distance between point P and any other point on the given line.

The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using the distance formula, we can calculate the distance between point P(2, -3) and the intersection point (5/2, -5/2):
d = sqrt((5/2 - 2)^2 + (-5/2 - (-3))^2)
d = sqrt((-1/2)^2 + (1/2)^2)
d = sqrt(1/4 + 1/4)
d = sqrt(1/2)
d = sqrt(2)/2

Now, let's calculate the distance between point P(2, -3) and any other point on the given line x + y = 0. Since the line passes through the origin (0, 0), the distance

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