1e:I[80107,["/_next/static/chunks/235ad55241b6443f.js","/_next/static/chunks/8337ab1f037b6654.js","/_next/static/chunks/c9e933b5fd1a0afc.js","/_next/static/chunks/848a330faa2461ac.js","/_next/static/chunks/9f2942e368a262d2.js","/_next/static/chunks/92f9e11244990470.js","/_next/static/chunks/a8b867fbb1dfca1a.js","/_next/static/chunks/1100f0390a901fa7.js","/_next/static/chunks/bbe6bd9d3c7b0355.js","/_next/static/chunks/a82195c3e5de7dd4.js"],"default"] 1f:I[76763,["/_next/static/chunks/235ad55241b6443f.js","/_next/static/chunks/8337ab1f037b6654.js","/_next/static/chunks/c9e933b5fd1a0afc.js","/_next/static/chunks/848a330faa2461ac.js","/_next/static/chunks/9f2942e368a262d2.js","/_next/static/chunks/92f9e11244990470.js","/_next/static/chunks/a8b867fbb1dfca1a.js","/_next/static/chunks/1100f0390a901fa7.js","/_next/static/chunks/bbe6bd9d3c7b0355.js","/_next/static/chunks/a82195c3e5de7dd4.js"],"default"] 1d:T438,{"@context":"https://schema.org","@graph":[{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://www.divinejee.com"},{"@type":"ListItem","position":4,"name":"Let lambda1 lambda2 be the values of lambda for which the po","item":"https://www.divinejee.com/question/let-lambda1-lambda2-be-the-values-of-lambda-for-which-the-points-frac-52-1-lambda-and-201-are"}]},{"@type":"QAPage","mainEntity":{"@type":"Question","name":"Question Let lambda1 lambda2 be the values of lambda for which the po on Topic","text":"

Let $\\lambda_{1}, \\lambda_{2}$ be the values of $\\lambda$ for which the points $\\left(\\frac{5}{2}, 1, \\lambda\\right)$ and $(-2,0,1)$ are at equal d...","answerCount":1,"upvoteCount":0,"datePublished":"2026-01-16","author":{"@type":"Organization","name":"DivineJEE"},"acceptedAnswer":{"@type":"Answer","text":"Check the detailed solution on the page.","upvoteCount":0,"url":"https://www.divinejee.com/question/let-lambda1-lambda2-be-the-values-of-lambda-for-which-the-points-frac-52-1-lambda-and-201-are"}}}]}20:T54e,Since $\left(\frac{5}{2}, 1, \lambda\right)$ and $(-2,0,1)$ are equidistant

from plane $2 x+3 y-6 z+7=0$

$$ \begin{aligned} & \therefore\left|\frac{2\left(\frac{5}{2}\right)+3(1)-6(\lambda)+7}{\sqrt{2^2+3^2+6^2}}\right|=\left|\frac{2(-2)+3(0)-6(1)+7}{\sqrt{2^2+3^2+6^2}}\right| \\\\ & \Rightarrow|5+3-6 \lambda+7|=|-4-6+7| \\\\ & \Rightarrow|15-6 \lambda|=|-3| \\\\ & \Rightarrow 15-6 \lambda= \pm 3 \\\\ & \Rightarrow 15-6 \lambda=3 \text { or } 15-6 \lambda=-3 \\\\ & \Rightarrow 6 \lambda=12 \text { or } 6 \lambda=18 \\\\ & \Rightarrow \lambda=2 \text { or } \lambda=3 \end{aligned} $$

$$ \begin{aligned} & \because \lambda_1>\lambda_2 \\\\ & \therefore \lambda_1=3 \text { and } \lambda_2=2 \end{aligned} $$

So, point will be $(1,2,3)$

Let $\mathrm{M}_0=(1,2,3)$

$M_1$ is the point through which line passes i.e, $(5,1,-7)$

and $\vec{s}=\hat{i}+2 \hat{j}+2 \hat{k}$

$$ \therefore \overrightarrow{\mathrm{M}_0 \mathrm{M}_1}=4 \hat{i}-\hat{j}-10 \hat{k} $$

Now, required distance $=\left|\frac{\overrightarrow{\mathrm{M}_0 \mathrm{M}_1} \times \vec{s}}{|\vec{s}|}\right|$

$$ \begin{aligned} & =\frac{|(4 \hat{i}-\hat{j}-10 \hat{k}) \times(\hat{i}+2 \hat{j}+2 \hat{k})|}{\sqrt{1+4+4}} \\\\ & =\frac{|18 \hat{i}-18 \hat{j}+9 \hat{k}|}{3}=9 \end{aligned} $$