The distance from the line x=2+t,y=1+t,z=−12−12t to the plane x+2y+6z=10 is λμ, then 5λ−μ is ________

The line is x−21=y−11=z+12−12=t......(i) Line passes through 2i∧+j∧−12k∧ and is parallel to the vector V→=i∧+j∧−12k∧ vector normal to the plane x+2y+6z=10 is n→=i∧+2j∧+6k∧ ∴V→.n→=1+2−3⇒ line (i) is parallel to the plane ∴d=|2+2−3−101+4+36|=941⇒λ=9,μ=41⇒5λ−μ=(5×9)−41=4

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

The distance from the line $x = 2 + t, y = 1 + t, z = - \frac{1}{2} - \frac{1}{2} t$ to the plane $x + 2 y + 6 z = 10$ is $\frac{λ}{\sqrt{μ}},$ then $5 λ - μ$ is ________

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

answer is 4.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

The line is $\frac{x - 2}{1} = \frac{y - 1}{1} = \frac{z + \frac{1}{2}}{- \frac{1}{2}} = t ...... (i)$
Line passes through $2 \overset{\land}{i} + \overset{\land}{j} - \frac{1}{2} \overset{\land}{k}$ and is parallel to the vector $\vec{V} = \overset{\land}{i} + \overset{\land}{j} - \frac{1}{2} \overset{\land}{k}$ vector normal to the plane $x + 2 y + 6 z = 10$ is $\vec{n} = \overset{\land}{i} + 2 \overset{\land}{j} + 6 \overset{\land}{k}$
$∴ \vec{V} . \vec{n} = 1 + 2 - 3 \Rightarrow$ line (i) is parallel to the plane
$∴ d = | \frac{2 + 2 - 3 - 10}{\sqrt{1 + 4 + 36}} | = \frac{9}{\sqrt{41}} \Rightarrow λ = 9, μ = 41 \Rightarrow 5 λ - μ = (5 \times 9) - 41 = 4$