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Given a straight line L which passes through the points P(2,-1) and Q(4,3) in R^2. Find the coordinates of the point S which is a reflection of the point R in the line L.

User Sixtyfive
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Final answer:

To find the coordinates of the point S which is a reflection of the point R in the line L, we can use the formula for reflection across a line. Given that the line L passes through the points P(2, -1) and Q(4, 3), the coordinates of point S are (3, 3).

Step-by-step explanation:

To find the coordinates of the point S which is a reflection of the point R in the line L, we can use the formula for reflection across a line. The formula is:

x' = x - 2 * (x * m + c)/(m^2 + 1)

y' = y - 2 * (y * m + c)/(m^2 + 1)

In this formula, (x, y) are the coordinates of the point R, and (x', y') are the coordinates of the reflected point S. Also, m is the slope of the line L and c is the y-intercept of the line L.

Given that the line L passes through the points P(2, -1) and Q(4, 3), we can first find the slope of the line L:

m = (3 - (-1))/(4 - 2) = 2/2 = 1

Next, we can find the y-intercept of the line L by substituting the coordinates of one of the points (let's use point P) into the equation of a line:

-1 = 2 * 1 + c

c = -3

Now we have the slope (m) and y-intercept (c) of the line L, so we can use the reflection formula to find the coordinates of the point S:

x' = x - 2 * (x * m + c)/(m^2 + 1)

y' = y - 2 * (y * m + c)/(m^2 + 1)

Substituting the coordinates of point R (x = 4, y = 3) into the reflection formula:

x' = 4 - 2 * (4 * 1 + (-3))/(1^2 + 1) = 4 - 2 * (4 - 3)/2 = 4 - 2/2 = 4 - 1 = 3

y' = 3 - 2 * (3 * 1 + (-3))/(1^2 + 1) = 3 - 2 * (3 - 3)/2 = 3 - 0 = 3

Therefore, the coordinates of point S are (3, 3).

User Anton Bessonov
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