The equation that represents the line k is y = x{(y₁ - y₂)/ x₁} + y₂
A reflection over line is a tranformation in which each point of the original figure (the preimage) has an image that is the same distance from the reflection line as the original point, but is on the opposite side of the line.
Now,the reflection across the y-axis of a line k gives an image that have the same y-intercept as the preimage.
Suppose,the coordinate of two points on the line are;
(x₁, y₁) and (x₂, y₂)
The line about which the line g is reflected is the line k
The coordinates of the image of the point (x, y) following a reflection about the y-axis is the point (x₁, y₁).
Therefore, the coordinates of the points on the line k are;
(-x₁, y₁) along y-axis is (x₁, y₁)
(0, y₂) along y-axis (-0, y₂) = (0, y₂)
The slope of the line m is therefore; y₁ - y₂/x₁-x₂ = y₁ - y₂/x₁
The equation of the line is therefore; y - y₁= m (x -x₁)
⇒y - y₁= y₁ - y₂/x₁*(x -x₁)
⇒(y - y₁)x₁ = (y₁ - y₂)*(x -x₁)
⇒ x₁ y - x₁y₁ = xy₁ - x y₂ - x₁y₁ + x₁ y₂
⇒ x₁ y = x(y₁ - y₂) + x₁ y₂
Which gives;
y = x{(y₁ - y₂)/ x₁} + y₂
Hence, the equation that represents the line k is y = x{(y₁ - y₂)/ x₁} + y₂
More about Reflection of line:
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