well srashti it isn't much tough , see there can be two line slopes equally inclined , use tan(45°)= m-m'/1+mm' find slope and use point(3,4) slope form form line equations and get their intersections and find the area. your welcome

To find the equations of the two lines inclined at 45° to the line x-y=2, we need to determine the slope of the given line and then calculate the slopes of the lines inclined at 45°.

Given line: x-y=2

To find the slope of this line, we can rewrite the equation in slope-intercept form which is y=mx+c, where m represents the slope. Rearranging the equation, we get:

y = x - 2

Comparing this with the slope-intercept form, we can see that the slope (m) of the given line is 1.

The lines inclined at 45° will have slopes that are negative reciprocals of each other. The negative reciprocal of 1 is -1. Therefore, the slopes of the two lines inclined at 45° are -1.

Using the point-slope form of a line, which is y - y1 = m(x - x1), we can find the equations of the two lines passing through the point (3,4) with a slope of -1.

Line 1:
Using the point (3,4), the equation of the line is:
y - 4 = -1(x - 3)
Simplifying this equation gives:
y - 4 = -x + 3
y = -x + 7

Line 2:
Using the point (3,4), the equation of the line is:
y - 4 = -1(x - 3)
Simplifying this equation gives:
y - 4 = -x + 3
y = -x + 7

Therefore, the equations of the two lines inclined at 45° to the line x-y=2 passing through the point (3,4) are y = -x + 7 for both lines.



To find the area of the triangle bounded by the three lines, we first need to find the points of intersection between the lines.

The given line x-y=2 intersects the lines y = -x + 7 at a point, which we can find by solving the two equations simultaneously.

x - y = 2
y = -x + 7

Substituting the second equation into the first equation, we have:
x - (-x + 7) = 2
x + x - 7 = 2
2x = 9
x = 9/2

Substituting the value of x back into either of the equations, we have:
y = -(9/2) + 7
y = 5/2

Therefore, the point of intersection between the given line and the lines inclined at 45° is (9/2, 5/2).

Now, we have three points: (3,4), (9/2, 5/2), and the point of intersection.

Using the distance formula, we can find the length of each side of the triangle formed by these points.

Side 1: Distance between (3,4) and (9/2, 5/2)
= sqrt((9/2 - 3)^2 + (5/2 - 4)^2)
= sqrt((9/2 - 6/2)^2