QJEEM23Ma190c: JEE Main 2023 Mathematics - Class 12 JEEM23Ma190c Solution: If the shortest between the lines x + 6 2 = y − 6 3 = z − 6 4…

Shortest distance between the lines

$\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$

$\frac{x-\lambda}{3}=\frac{y-2 \sqrt{6}}{4}=\frac{2+2 \sqrt{6}}{5}$ is 6

Vector along line of shortest distance

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padding: 0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0.5em; padding-top: 0.2em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: left; padding-right: 0.5em; padding-top: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0.5em; padding-right: 0.5em; padding-top: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0.5em; padding-top: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="left left left" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">i</mi></mrow></mtd><mtd><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">j</mi></mrow></mtd><mtd><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">k</mi></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container>\right|, \Rightarrow-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\mathrm{k}<span>(its magnitude is$</span>\sqrt{6}$ )

Now \frac{1}{\sqrt{6}}\left|<mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="4" style="font-size: 130.9%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mtable style="min-width: 9.142em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right: 0.5em; padding-bottom: 0.2em;"><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top: 0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D706 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-right: 0.5em; padding-bottom: 0.2em;"><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top: 0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; 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$\Rightarrow \lambda=-2 \sqrt{6}, 10 \sqrt{6}$

So, square of sum of these values

$$=(10\sqrt{6}-2\sqrt{6}{)}^2=(8\sqrt{6}{)}^2=384$$