Go To Your Study Dashboard

Get Answers to all your Questions

header-bg

Let $\mathrm{\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z+3}{-1}}$ lie on the plane $\mathrm{p x-q y+z=5}$ for some $\mathrm{p, q \in \mathbb{R}}$ . The shortest distance of the plane from the origin is:
Option: 1
$\mathrm{ \sqrt{\frac{3}{109}} \\ }$

Option: 2
$\mathrm{\sqrt{\frac{5}{142}} \\ }$

Option: 3
$\mathrm{\frac{5}{\sqrt{71}} }$

Option: 4
$\mathrm{\frac{1}{\sqrt{142}}}$

Answers (1)

best_answer

For line to lie on the plane, point $\mathrm{A(2,-1,-3)}$ lying on line should lie on plane as well

$\mathrm{\therefore \quad 2 p+q-3=5 \Rightarrow 2 p+q=8}$ .........(i)

Also vector parallel to line $\mathrm{(3i-2j-k)}$ should be perpendicular to normal vector $\vec{n}$ of plane $\mathrm{\left ( pi-qj+k \right )}$

$\mathrm{\therefore (3 i-2 j-k) \cdot(p i-q j+k)=0} \\$

$\mathrm{\Rightarrow 3 p+2 q-1=0} \\$

$\mathrm{\Rightarrow 3 p+2 q=1 \quad \ldots . \text { (ii) }}$

From (i) and (ii)

$\mathrm{p=15, q=-22}$

$\mathrm{\therefore \text { plane is } 15 p+22 y+z-5=0}$

Distance from origin $\mathrm{(0,0,0)}$

$\mathrm{=\frac{|-5|}{\sqrt{15^{2}+22^{2}+1^{2}}}} \\$

$\mathrm{=\frac{5}{\sqrt{710}}} \\$

$\mathrm{=\sqrt{\frac{5}{142}}}$

hence answer is option 2

Posted by

qnaprep

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

5 g of Na₂SO₄ was dissolved in x g of H₂O. The change in freezing point was found to be 3.82⁰C. If Na₂SO₄ is 81.5% ionised, the value of x (K

A capacitor is made of two square plates each of side 'a' making a very small angle $\alpha$

A solution of m-chloroaniline, m-chlorophenol and m-chlorobenzoic acid in ethyl acetate was extracted initially with a saturated solution of NaHCO₃ to give fraction A. The leftover organic phase was extracted with d

Trending Articles/News

anam-khan

Engineering aspirant dies by suicide in Kota, fourth case this year

anam-khan

What is non-Aadhaar declaration in JEE Main 2025? Explain mismatch between image, face

anam-khan

Forgot JEE Main password 2025 for downloading admit card? Here’s what to do

Latest Question