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:T441,{"@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 O be the origin and M and N be the points on the lines f","item":"https://www.divinejee.com/question/let-o-be-the-origin-and-m-and-n-be-the-points-on-the-lines-frac-x54frac-y41frac-z53-and-frac-x-812"}]},{"@type":"QAPage","mainEntity":{"@type":"Question","name":"Question Let O be the origin and M and N be the points on the lines f on Topic","text":"

Let O be the origin, and M and $\\mathrm{N}$ be the points on the lines $\\frac{x-5}{4}=\\frac{y-4}{1}=\\frac{z-5}{3}$ and $\\frac{x+8}{12}=\\frac{y+2}{5...","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-o-be-the-origin-and-m-and-n-be-the-points-on-the-lines-frac-x54frac-y41frac-z53-and-frac-x-812"}}}]}21:T507,

$$\mathrm{L}_1: \frac{\mathrm{x}-5}{4}=\frac{\mathrm{y}-4}{1}=\frac{\mathrm{z}-5}{3}=\lambda \quad \operatorname{drs}(4,1,3)=\mathrm{b}_1$$

$$\begin{aligned} & \mathrm{M}(4 \lambda+5, \lambda+4,3 \lambda+5) \\ & \mathrm{L}_2: \frac{\mathrm{x}+8}{12}=\frac{\mathrm{y}+2}{5}=\frac{\mathrm{z}+11}{9}=\mu \\ & \mathrm{N}(12 \mu-8,5 \mu-2,9 \mu-11) \\ & \overrightarrow{\mathrm{MN}}=(4 \lambda-12 \mu+13, \lambda-5 \mu+6,3 \lambda-9 \mu+16) \quad \text{..... (1)} \end{aligned}$$

Now

$$\overrightarrow{\mathrm{b}}_1 \times \overrightarrow{\mathrm{b}}_2=\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 4 & 1 & 3 \\ 12 & 5 & 9 \end{array}\right|=-6 \hat{\mathrm{i}}+8 \hat{\mathrm{k}} \quad \text{.... (2)}$$

Equation (1) and (2)

$$\therefore \frac{4 \lambda-12 \mu+13}{-6}=\frac{\lambda-5 \mu+6}{0}=\frac{3 \lambda-9 \mu+16}{8}$$

I and II

$$\lambda-5 \mu+6=0 \quad \text{.... (3)}$$

I and III

$$\lambda-3 \mu+4=0 \quad \text{.... (4)}$$

Solve (3) and (4) we get

$$\begin{aligned} \lambda= & -1, \mu=1 \\ \therefore \quad & \mathrm{M}(1,3,2) \\ & \mathrm{N}(4,3,-2) \\ \therefore \quad & \overrightarrow{\mathrm{OM}} \cdot \overrightarrow{\mathrm{ON}}=4+9-4=9 \end{aligned}$$