
Find the image of the point (3,8) about the line x+3y = 7 assuming the line to be a plane mirror.
Answer
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Hint: Assume the coordinates of the reflection point be B(x,y). Use the fact that the image distance is equal to object distance. Use the fact that the line joining object to the image is orthogonal to the reflecting surface. Hence form two equations in x and y. Solve for x and y. Hence find coordinates of B.
Alternatively, use the fact that the image of in the line mirror is given by
, where h,k are the coordinates of the image.
Complete step-by-step solution -
Let the coordinates of the image be B(x,y) and let the line AB intersect the line mirror at D.
Since image distance is equal to object distance, we have AD = DB
Hence
Hence D is the midpoint of AB.
Now we know that coordinates of the midpoint of AB, where and , are given by
Here and
Hence
Since D lies on the line mirror, we have
.
Also, AB and the line mirror are perpendicular to each other.
Slope of AB
Slope of mirror
We know that if the slopes of two perpendicular lines are and then
Hence we have
Substituting the value of k from equation (ii) in equation (i), we get
Substituting the value of h in equation (ii), we get
Hence the coordinates of the image are given by (-1,-4).
Note: Alternative Solution:
We know that the image of in the line mirror is given by
, where h,k are the coordinates of the image.
Here and
Hence we have
Hence and
Hence the coordinates of the image are given by
(-1,-4), which is the same as obtained above.
Alternatively, use the fact that the image of
Complete step-by-step solution -

Let the coordinates of the image be B(x,y) and let the line AB intersect the line mirror at D.
Since image distance is equal to object distance, we have AD = DB
Hence
Hence D is the midpoint of AB.
Now we know that coordinates of the midpoint of AB, where
Here
Hence
Since D lies on the line mirror, we have
Also, AB and the line mirror are perpendicular to each other.
Slope of AB
Slope of mirror
We know that if the slopes of two perpendicular lines are
Hence we have
Substituting the value of k from equation (ii) in equation (i), we get
Substituting the value of h in equation (ii), we get
Hence the coordinates of the image are given by (-1,-4).
Note: Alternative Solution:
We know that the image of
Here
Hence we have
Hence
Hence the coordinates of the image are given by
(-1,-4), which is the same as obtained above.
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