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 coordinate
axes a triangle of area S. If $α β > 0$ , then the least value of S is

Moderate

A

$α β$
B

$2 α β$
C

$4 α β$
D

none of these

Solution

The equation of the given line is
$\frac{x}{a} + \frac{y}{b} = 1$ (1)
This line cuts x-axis and y-axis at A(a, 0) and B(0, b) respectively.
Since area of $∆$ OAB = S (Given)
$∴ |\frac{1}{2} a b| = S o r a b = 2 S (∵ a b > 0)$ (2)
Since the line (1) passes through the point $P (α, β)$
$∴ \frac{α}{a} + \frac{β}{b} = 1 o r \frac{α}{a} + \frac{α β}{2 S} = 1 [U \sin g (2)]$
$o r α^{2} β - 2 a S + 2 α S = 0$
Since a is real, $∴ 4 S^{2} - 8 α β S \geq 0$
$o r 4 S^{2} \geq 8 α β S o r S \geq 2 α β (∵ S = \frac{1}{2} a b > 0 a s a b > 0)$
Hence the least value of $S = 2 α β$

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