Let l be the line whose question is 2x-y=10.which one of the following is equation of the image of after reflection in the line y=2x-5 followed by a rotation through the angle of 90 about origin? ?

To find the equation of the image of line l after reflection in the line y=2x-5 followed by a rotation of 90 degrees about the origin, we can follow these steps:

1. Find the reflection of line l in the line y=2x-5:
The reflection of a point (x, y) in the line y=2x-5 can be found by using the formula:
(x', y') = (x - 2(y-2x-5)/(1+2^2), y - 2(x-2y+5)/(1+2^2))

Substitute x=2, y=10 into the formula:
(x', y') = (2 - 2(10-2*2-5)/(1+2^2), 10 - 2(2-2*10+5)/(1+2^2))
(x', y') = (2 - 2(10-4-5)/5, 10 - 2(2-20+5)/5)
(x', y') = (2 - 2(1)/5, 10 - 2(-13)/5)
(x', y') = (2 - 2/5, 10 + 26/5)
(x', y') = (8/5, 58/5)

So, the reflection of line l in the line y=2x-5 is the point (8/5, 58/5).

2. Rotate the point (8/5, 58/5) by 90 degrees about the origin:
The rotation of a point (x, y) by 90 degrees about the origin can be found by using the formula:
(x', y') = (xcos(90) - ysin(90), xsin(90) + ycos(90))

Substitute x=8/5, y=58/5 into the formula:
(x', y') = (8/5*0 - 58/5*1, 8/5*1 + 58/5*0)
(x', y') = (-58/5, 8/5)

Therefore, the equation of the image of line l after reflection in the line y=2x-5 followed by a rotation of 90 degrees about the origin is:
y = -58/5x + 8/5

The equation of a line is given by y = mx + c, where m is the slope and c is the y-intercept.
In this case, the given line is 2x - y = 10, which can be rewritten as y = 2x - 10.
This implies the slope of the given line is 2 and y-intercept is -10.

The reflection of the given line about the line y = 2x - 5 means the perpendicular distance from any point on the reflected line to the reflection axis y = 2x - 5 is the same as the perpendicular distance from the corresponding point on the original line to the reflection axis y = 2x - 5.
The perpendicular distance of any point (x, y) on the line y = 2x - 10 to a line y = mx + c is given by d = |(y - mx - c) / sqrt(m^2 + 1)|.

Substituting y = 2x - 10 and m = 2 and c = -5 in the above equation, we get d = |(2x - 10 - 2x + 5) / sqrt(2^2 + 1)| = |5 / sqrt(5)| = sqrt(5).

So, the perpendicular distance from any point on the reflected line to the reflection axis y = 2x - 5 is also sqrt(5).

The equation of the reflected line is y = 2x - c', where c' is the y-intercept of the reflected line.
The y-intercept of the reflected line is the point where the line intersects the y-axis.
To find the y-intercept of the reflected line, we substitute x = 0 in the equation of the reflected line: y = 2(0) - c' = -c'.
So, the y-intercept of the reflected line is -c'.

The reflected line is parallel to the given line. Thus the slope of reflected line is also 2.

The rotation of the reflected line about the origin through an angle of 90 degrees means the x and y coordinates of each point on the rotated line are interchanged.
So, the equation of the image of the given line after reflection in the line y = 2x - 5 followed by a rotation through an angle of 90 degrees about the origin is x = 2y - c'', where c'' is the x-intercept of the rotated line.

To find the equation of the rotated line we need to find the value of c''.
We can find c'' by substituting x = 0 in the equation x = 2y - c'': 0 = 2(0) - c'' => c'' = 0.

Thus the equation of the image of the given line after reflection in the line y = 2x - 5 followed by a rotation through an angle of 90 degrees about the origin is x = 2y.