Answer :

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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