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Reflection in Lines Parallel to Axes Examples
Example 1.
The reflection of the point (4, -1) about the line 5x + y + 6 = 0 is
Solution:
Given that
The reflection point be A (h, k)
Now, the mid point of the line joining (h, k) and (4, -1) will lie on the line
5x + y + 6 = 0
Therefore
5(h+4)/2 + (k−1)/2 + 6 = 0
5h + 20 + k – 1 + 12 = 0
5h + k + 31 = 0……(i)
Now, the slope of the line joining points (h, k) and (4, -1) are perpendicular to the line is
5x + y + 6 = 0
Slope of the line=−5
Slope of the joining by points (h, k) and (4, -1)
k+1/h−4
Therefore
k+1/h−4(−5) = -1
5k − h + 69 = 0…..(ii)
Solving (i) and (ii), we get
5h + k + 6 – 5k + h – 69 = 0
6h – 4k – 63
h = 6 and k = -4
Example 2.
The point (-3, 0) on reflection in a line is mapped to (3, 0) and the point (4, -6) on reflection in the same line is mapped to (-4, -6).
(i) State the name of the mirror line and write its equation.
(ii) State the coordinates of the image of (-7, -5) in the mirror line.
Solution:
(i) We know the reflection of a point (x, y) in the y-axis is (-x, y).
Hence, the point (-3, 0) when reflected in the y-axis is mapped to (3, 0).
Therefore, the mirror line is the y-axis and its equation is x = 0.
(ii) Coordinates of the image of (-7, -5) in the mirror line in the y-axis are (7, -5).
Example 3.
The point (-4, 0) on reflection in a line is mapped as (4, 0) and the point (-7, -6) on reflection in the same line is mapped as (2, -6).
(a) Name the line of reflection.
(b) Write the coordinates of the image of (5, -9).
Solution:
(a) We know that reflection in line x = 0 is the reflection in the y-axis.
Given that the point is
Point (-4, 0) on reflection in a line is mapped as (4, 0).
Point (-7, -6) on reflection in the same line is mapped as (7, -6).
Hence, the line of reflection is x = 0.
(b) we know that
My (x, y) = (-x, y)
Coordinates of the image of (5, -9) in the line x = 0 are (-5, -9).
Example 4.
Point P(4, -3) is reflected as P’ in the y-axis. Point B on reflection in the x-axis is mapped as Q’ (-2, 7). Write the coordinates of P’ and Q.
Solution:
Given that the points are P(4,-3) and Q(-2,7)
Reflection in y-axis is given by
My (x, y) = (-x, y)
P’ = Reflection of P(4, -3) in y-axis
P’= (-4, -3)
Reflection in x-axis is given by
Mx (x, y) = (x, -y)
Q’ = Reflection of Q in x-axis = (-2, 7)
Therefore Q = (-2, -7)
Example 5.
The point (-3, 0) on reflection in a line is mapped as (3, 0) and the point (-8, -6) on reflection in the same line is mapped as (4, -6).
(a) Name the line of reflection.
(b) Write the coordinates of the image of (5, -6).
Solution:
(a) We know that reflection in line x = 0 is the reflection in the y-axis.
Given that the point is
Point (-3, 0) on reflection in a line is mapped as (3, 0).
Point (-8, -6) on reflection in the same line is mapped as (8, -6).
Hence, the line of reflection is x = 0.
(b) we know that
My (x, y) = (-x, y)
Coordinates of the image of (5, -6) in the line x = 0 are (-5, -6).
FAQs on Reflection in Lines Parallel to Axes
1. What does parallel to the axes mean?
If a line is parallel to the x-axis or y-axis either the x-coordinate or y-coordinate is constant or fixed throughout the line and it should pass through either (0, a) or (a, 0).
2. How do you reflect over the axis?
A reflection of a point over the y -axis is. The rule for a reflection over the y -axis is (x,y) = (-x,y). And the reflection over the x- axis is (x,y) = (x,-y).
3. What Does Parallel to the Axes Mean?
Parallel to axes means that the lines are parallel to either the x-axis or y-axis.
If a line parallel to the x-axis is called a horizontal line whose equation is in the form of y = k,
where ‘k’ is the distance of the line from the x-axis.
Similarly, a line parallel to the y-axis is called a vertical line whose equation is in the form of x = k,
where ‘k’ is the distance of the line from the y-axis.