The value of maximum area possible of a DABC such that A(0, 0) and B(x, y) and C(–x, y) such that y = –2x² + 54x is (in sq. unit)

Correct Option - 2


Detailed Solution:

Area of ?

$=\frac{1}{2}\left|\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill x\hfill & \hfill y\hfill & \hfill 1\hfill \\ \hfill -x\hfill & \hfill y\hfill & \hfill 1\hfill \end{array}\right|$

-> $\left|\frac{1}{2}\left(xy+xy\right)\right|=\left|xy\right|$

->Area (D) = |xy| = |x (– 2x² + 54x)|

$\frac{d\left(\Delta \right)}{dx}=\left|\left(-6x^2+108x\right)\right|\Rightarrow \frac{d\Delta }{dx}=0$  at x = 0 and 18

->at x = 0, minima

and at x = 18 maxima

Area (D) = |18 (– 2 (18)² + 54 × 18)| = 5832

<div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1759124921phpuBi3ga.jpeg" alt="" width="307" height="196" loading="lazy"></picture></div></div><p><span lang="EN-IN">Area of </span><span lang="EN-IN">? </span></p><p><span contenteditable="false"> <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>|</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mi>x</mi> </mtd> <mtd columnalign="center"> <mi>y</mi> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mo>−</mo> <mi>x</mi> </mtd> <mtd columnalign="center"> <mi>y</mi> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mrow> </math> </span></p><p><span lang="EN-IN">-&gt;<span contenteditable="false"> <math> <mrow> <mrow> <mo>|</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo> (</mo> <mrow> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>|</mo> </mrow> </mrow> </math> </span>&lt;!- [endif]-&gt;&lt;!- [if gte mso 9]-&gt; &lt;!- [endif]-&gt;</span></p><p><span lang="EN-IN">-&gt;Area (</span><span lang="EN-IN">D</span><span lang="EN-IN">) = |xy| = |x (– 2x² + 54x)|</span></p><p><span lang="EN-IN"><span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mi>d</mi> <mrow> <mo> (</mo> <mrow> <mi>Δ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mrow> <mo> (</mo> <mrow> <mo>−</mo> <mn>6</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mn>0</mn> <mn>8</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>⇒</mo> <mfrac> <mrow> <mi>d</mi> <mi>Δ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </mrow> </math> </span>&nbsp;at x = 0 and 18</span></p><p><span lang="EN-IN">-&gt;at x = 0, minima</span></p><p><span lang="EN-IN">and at x = 18 maxima</span></p><p><span lang="EN-IN">Area (</span><span lang="EN-IN">D</span><span lang="EN-IN">) = |18 (– 2 (18)² + 54 × 18)| = 5832</span></p>

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