First slide

Cartesian plane

Question

$The line \frac{x}{a} + \frac{y}{b} = 1 meets the x -axis at A and y -axis at B and the line y = x at C such that the area of the △ A O C is$
$twice the area of Δ B O C . Then the coordinates of C are$

Easy

A

$(\frac{b}{3}, \frac{b}{3})$
B

$(\frac{2 a}{3}, \frac{2 a}{3})$
C

$(\frac{2 b}{3}, \frac{2 b}{3})$
D

None of these

Solution

$\begin{aligned} Given Δ_{A O C} = 2 (Δ_{B O C}) \\ \Rightarrow \frac{1}{2} (O A) (x_{1}) = \frac{2 \times 1}{2} (O B) (x_{1}) \\ \Rightarrow a = 2 b \end{aligned}$
$Equation of A B \Rightarrow \frac{x}{a} + \frac{y}{b} = 1 - - - - (1)$
$\frac{x}{2 b} + \frac{y}{b} = 1 - - - - (2)$
$\begin{array}{l} \Rightarrow Since point C lies on the line (2) \\ \Rightarrow \frac{x_{1}}{2 b} + \frac{x_{1}}{b} = 1 \\ \Rightarrow x_{1} = \frac{2 b}{3} = \frac{a}{3} \\ \Rightarrow C = (\frac{2 b}{3}, \frac{2 b}{3}) \end{array}$

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