First slide

Equation of line in Straight lines

Question

$Through the point P (α, β), where α β > 0, the straight line \frac{x}{a} + \frac{y}{b} = 1 is drawn so as to form with axes a triangle of$
$area S . If a b > 0, then least value of S is$

Moderate

A

$α β$
B

$2 α β$
C

$3 α β$
D

None of these

Solution

$Area of Δ O A B = S = \frac{1}{2} a b - - - - (1)$
$Equation of A B is \frac{x}{a} + \frac{y}{b} = 1$
Putting $(α, β)$ , we get
$\frac{α}{a} + \frac{β}{b} = 1$
from (1) $b = \frac{2 S}{a}$
$\Rightarrow \frac{α}{a} + \frac{a β}{2 S} = 1 - - - - - - (2)$
$\begin{array}{l} \Rightarrow a^{2} β - 2 a S + 2 α S = 0 \\ ∴ a \in R \\ \Rightarrow D \geq 0 \\ 4 S^{2} - 8 α β S \geq 0 \\ S \geq 2 α β \\ Least value of S = 2 α β \end{array}$

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