# The line 2x + 3y = 12 meets the x-axis at A and y-axis at B. The line through (5, 5) perpendicular to AB meets the x-axis and the line AB at C and E respectively. If O is the origin of coordinates, find the area of figure OCEB.

Given:

Line 2x + 3y = 12 meets the x-axis at A and y-axis at B

To find:

The area of figure OCEB.

Explanation:

The given line is 2x + 3y = 12, which can be written as

……(1)

So, the coordinates of the points A and B are (6, 0) and (0, 4), respectively.

Diagram:

The equation of the line perpendicular to line (1) is

This line passes through the point (5, 5).

Now, substituting the value of λ in , we get:

…….(2)

Thus, the coordinates of intersection of line (1) with the x-axis is C

To find the coordinates of E, let us write down equations (1) and (2) in the following manner:

2x + 3y – 12 = 0 … (3)

3x – 2y – 5 = 0 … (4)

Solving (3) and (4) by cross multiplication, we get:

x = 3, y = 2

Thus, the coordinates of E are (3, 2).

From the figure,

EC

EA

Now,

Area(OCEB) = Area(OAB) –Area(CAE)

Area(OCEB)

sq units

Hence, area of figure OCEB is sq units

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