Find the area of the triangle formed by the line y - x = 0, x + y = 0 and x - k = 0

Solution:

The equations of the given lines are:

y - x = 0 ....(1)

x + y = 0 ....(2)

x - k = 0 ....(3)

The point of intersection of lines (1) and (2) is obtained by solving them.

x = 0 and y = 0

The point of intersection of lines (2) and (3) is obtained by solving them.

x = k and y = - k.

The point of intersection of lines (3) and (1) is obtained by solving them.

x = k and y = k.

Thus, the vertices of the triangle formed by the three given lines are (0, 0), (k, - k) and (k, k).

We know that the area of a triangle whose vertices are (x\(_1\), y\(_1\)), (x\(_2\), y\(_2\)) and (x\(_3\), y\(_3\)) is

1/2 |x\(_1\) (y\(_2\) - y\(_3\)) + x\(_2\) (y\(_3\) - y\(_1\)) + x\(_3\) (y\(_1\) - y\(_2\))|

Therefore, area of the triangle formed by three given lines

= 1/2 |0 (- k + k) + k (k - 0) + k (0 + k)|

= 1/2 |k² + k²|

= 1/2 (2k²)

= k²

Hence, the area of the triangle is k² square units

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NCERT Solutions Class 11 Maths Chapter 10 Exercise ME Question 8

Find the area of the triangle formed by the line y - x = 0, x + y = 0 and x - k = 0

Summary:

The area of the triangle formed by the line y - x = 0, x + y = 0 and x - k = 0 is k² square units