1f: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"] 20: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"] 1e:T43d,{"@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 P1 be the plane 3 xy7 z11 and P2 be the plane passing th","item":"https://www.divinejee.com/question/let-p1-be-the-plane-3-xy7-z11-and-p2-be-the-plane-passing-through-the-points-210201-and-511-if-the"}]},{"@type":"QAPage","mainEntity":{"@type":"Question","name":"Question Let P1 be the plane 3 xy7 z11 and P2 be the plane passing th on Topic","text":"

Let $\\mathrm{P}_{1}$ be the plane $3 x-y-7 z=11$ and $\\mathrm{P}_{2}$ be the plane passing through the points $(2,-1,0),(2,0,-1)$, and $(5,1,1)$. I...","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-p1-be-the-plane-3-xy7-z11-and-p2-be-the-plane-passing-through-the-points-210201-and-511-if-the"}}}]}21:T547,Equation of plane $\mathrm{P}_2$ passing through $(2,-1,0),(2,0$, $-1)$ and $(5,1,1)$ is

$$ \begin{aligned} & \left|\begin{array}{ccc} x-5 & y-1 & z-1 \\ 3 & 2 & 1 \\ 3 & 1 & 2 \end{array}\right|=0 \\\\ & \Rightarrow(x-5)(4-1)-(y-1)(6-3)+(z-1)(3-6)=0 \\\\ & \Rightarrow 3 x-15-3 y+3-3 z+3=0 \\\\ & \Rightarrow 3 x-3 y-3 z-9=0 \\\\ & \Rightarrow x-y-z=3 .......(i) \end{aligned} $$

Now, direction ratios of line of intersection of $P_1$ and $\mathrm{P}_2$ is

$$ \begin{aligned} & \left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & -1 \\ 3 & -1 & -7 \end{array}\right| \\\\ & =\hat{i}(7-1)-\hat{j}(-7+3)+\hat{k}(-1+3) \\\\ & =6 \hat{i}+4 \hat{j}+2 \hat{k} \end{aligned} $$

At $z=0, x-y=3$ [from (i)]

$$ 3 x-y=11 $$

On solving, we get

$$ x=4 \text { and } y=1 $$

So, equation of line is

$$ \frac{x-4}{6}=\frac{y-1}{4}=\frac{z-2}{6}=k $$

$$ \begin{aligned} & \therefore(\alpha, \beta, \gamma)=(6 k+4,4 k+1,2 k) \\\\ & \Rightarrow(6)(\alpha-7)+4(\beta-4)+2(\gamma+1)=0 \\\\ & \Rightarrow 6(6 k+4-7)+4(4 k+1-4)+2(2 k+1)=0 \\\\ & \Rightarrow 36 k-18+16 k-12+4 k+4=0 \\\\ & \Rightarrow 56 k=26 \Rightarrow k=\frac{1}{2} \\\\ & \text { So, } \alpha=7, \beta=3 \text { and } \gamma=1 \\\\ & \therefore \alpha+\beta+\gamma=7+3+1=11 \end{aligned} $$