Q. 243.7( 3 Votes )
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.
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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