The line 2 x+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.
The given line is 2x+3y=12, which can be written as
So, the coordinates of the points A and B are (6, 0) and (0, 4) respectively.
The equation of the line perpendicular to line (i) is
This line passes through the point (5, 5)
Now, substituting the value of λ in x4−y6+λ=0, we get:
⇒ x52−y52=1 ...(ii)
Thus, the coordinates of intersection of line (i) with the x-axis is C (53,0)
To find the coordinates of E, let us write down equations (i) and (ii) in the following manner:
Solving (iii) and (iv) by cross multiplication, we get:
⇒ x=3, y=2
Thus, the coordinates of E are (3, 2) From the figure,
Area (OCEB) = Area (Δ OAB) - Area (ΔCAE)