A line and area of a triangle

Here's a statement Area of triangle formed by the line passing through the point $\left( \alpha ,\beta \right)$ , with axes is maximum when point $\left( \alpha ,\beta \right)$ is the midpoint of the line between the axes.

The question was to answer whether the statement was correct or not and the answer given is that the statement is WRONG.

But why?

Here's what I did,

Let $A(0,n)$ , $B(\alpha,\beta)$ , $C(h,0)$ and $AB:BC::k:1$

We know, $\alpha =\frac { kh }{ k+1 } \quad \& \quad \beta =\frac { n }{ k+1 }$

$\Rightarrow h=\frac { \alpha \left( k+1 \right) }{ k } \quad \& \quad n=\beta \left( k+1 \right)$

Area of $\Delta AOC$ , $A=\frac { 1 }{ 2 } \times \frac { \alpha \left( k+1 \right) }{ k } \times \beta \left( k+1 \right)$

Differentiating w.r.t k,

$\Rightarrow \frac { dA }{ dk } =\frac { 1 }{ 2 } \alpha \beta \left[ \frac { 2k\left( k+1 \right) -{ \left( k+1 \right) }^{ 2 } }{ { k }^{ 2 } } \right]$

Equating $\frac { dA }{ dk }$ to zero $\Rightarrow k=\pm 1$

Maxima occurs at $k=1$

$\therefore$ Area is maximum when $k=1$ .

$k=1\Rightarrow$ $(\alpha,\beta)$ is the midpoint of $A$ and $C$ .

Hence, Statement is CORRECT.

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

An application of AM-GM will tell you why you've gotten the wrong answer.

Note that $\dfrac{x}{a} + \dfrac{y}{b} = 1$ is the equation of the line in intercept form. $a$ and $b$ are the $x$ and $y$ intercepts respectively.
We need to maximize $\triangle = \dfrac{ab}{2}$ .
As $\alpha, \beta$ lies on the line, $\dfrac{\alpha}{a} + \dfrac{\beta}{b} = 1 \geq 2 \sqrt{\dfrac{\alpha \beta}{2 \triangle}}$
$\Rightarrow \triangle \geq 2 \alpha \beta$ with equality when $\dfrac{\alpha}{a} = \dfrac{\beta}{b} = \dfrac{1}{2}$

So you have the minimum area, not the maximum.
Be careful when using calculus to find extrema - check minima and maxima always.

Ameya Daigavane - 5 years, 2 months ago

Oh! yes sir, I made a mistake while checking maxima and minima! And your approach is superb!

The red curve represent $\frac { dA }{ dk }$ and green curve represent $\frac { { d }^{ 2 }A }{ d{ k }^{ 2 } }$

Anandhu Raj - 5 years, 2 months ago

@Ameya Daigavane Sorry disturb you again, sir. But could you please post a solution to below question?

A variable line, drawn through the point of intersection of the straight lines $\frac { x }{ a } +\frac { y }{ b } =1$ and $\frac { x }{ b } +\frac { y }{ a } =1$ , meets the coordinate axes in $A$ and $B$ . Then find the locus of the midpoint of $AB$ .

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$