A line with positive direction cosines passes through the point P… - Vidyadip

MCQ

A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the co-ordinate axes. The line meets the plane $2 x+y+z=9$ at point $Q$. The length of the line segment $P Q$ equals

A
3
B
$\sqrt{2}$
C
$\sqrt{3}$
D
2

✓

Answer

(c) : Since, line makes equal angles with coordinate axes, so direction cosine of line are $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$ Any point $Q$ on the line at a distance $t$ from $P(2,-1,2)$ is given by $\left(2+\frac{t}{\sqrt{3}},-1+\frac{t}{\sqrt{3}}, 2+\frac{t}{\sqrt{3}}\right)$
This point $Q$ meets the plane $2 x+y+z=9$.
$\begin{aligned}
& \Rightarrow 2\left(2+\frac{t}{\sqrt{3}}\right)+\left(-1+\frac{t}{\sqrt{3}}\right)+\left(2+\frac{t}{\sqrt{3}}\right)=9 \\
& \Rightarrow 4+\frac{2 t}{\sqrt{3}}-1+\frac{t}{\sqrt{3}}+2+\frac{t}{\sqrt{3}}=9 \Rightarrow 5+\frac{4 t}{\sqrt{3}}=9 \\
& \Rightarrow \frac{t}{\sqrt{3}}=1 \Rightarrow t=\sqrt{3}
\end{aligned}$
$\therefore \quad$ Distance between $P$ and $Q$ is $\sqrt{3}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Line and Plane→Line and Plane — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

The equation $2 x^2-4 x y- p y^2+4 x+ q y+1=0$ will represent two mutually perpendicular straight lines, if

If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are non$-$coplanar vectors, then $\frac{\vec{\text{a}}.\big(\vec{\text{b}}\times\vec{\text{c}}\big)}{\big(\vec{\text{c}}\times\vec{\text{a}}\big).\vec{\text{b}}}+\frac{\vec{\text{b}}.\big(\vec{\text{a}}\times\vec{\text{c}}\big)}{\vec{\text{c}}.\big(\vec{\text{a}}\times\vec{\text{b}}\big)}$ is equal to:

If the slope of one of the two lines $\frac{x^2}{a}+\frac{2 x y}{h}+\frac{y^2}{h}=0$ is twice that of the other, then ab:h ${ }^2$$=\ldots$.

The value of $\int_0^1 \frac{x^4(1-x)^4}{1+x^2} d x$ is

The combined equation of the co-ordinate axes is _________.

If $A=\left(\begin{array}{rr}\lambda & 1 \\ -1 & -\lambda\end{array}\right)$, then $A^{-1}$ does not exist if $\lambda=$

If sum of two unit vectors is itself a unit vector, then the magnitude of their difference is

$\int(\text{x}-1)\text{e}^{-\text{x}}\text{ dx}$ is equal to:

Let $f : R \rightarrow R$ be a function defined by $\text{f(x)}=\frac{\text{e}^{|\text{x}|}-\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}.$ Then,

The solution of the differential equation $\frac{\text{dy}}{\text{dx}}-\text{Ky}=0, \text{y}(0)=1$ approaches to zero when $\text{x}\rightarrow\propto$ if,