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JEE Advanced 2009Paper 2

Exam: JEE Advanced
Year: 2009
Paper: Paper 2
Subject: Mathematics
Chapter: Three Dimensional Geometry

JEE Advanced 2009 · Paper 2 · Mathematics · Three Dimensional Geometry

A line with positive direction cosines passes through the point...

A line with positive direction cosines passes through the point

P(2,-1,2)

and makes equal angles with the coordinate axes. The line meets the plane

2 x+y+z=9

at point

Q

. The length of the line segment

P Q

equals

A line with positive direction cosines passes through the point P(2,-1,2) and makes equal angles with the coordinate

A line with positive direction cosines passes through the point P(2,-1,2) and makes equal angles with the coordinate

Options

A1
B $\sqrt{2}$
C $\sqrt{3}$
D2

Correct answer

C. $\sqrt{3}$

Step-by-step solution

\text{ Since, }

l=m=n=

\frac{1}{\sqrt{3}}<img src="/api/proxy-image?url=https%3A%2F%2Ffirebasestorage.googleapis.com%2Fv0%2Fb%2Fquantrexacademy-app.firebasestorage.app%2Fo%2Fquestions%252Ffigs%252Fpyq%252Fjee_advanced%252Fich1orC4Q3FFYIKtKIixDQa3M4swfCBvT6uaMCex8bo.original.fullsize.png%3Falt%3Dmedia&clean=1&v=wm2" data-qx-orig-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_advanced%2Fich1orC4Q3FFYIKtKIixDQa3M4swfCBvT6uaMCex8bo.original.fullsize.png?alt=media" data-qx-storage-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_advanced%2Fich1orC4Q3FFYIKtKIixDQa3M4swfCBvT6uaMCex8bo.original.fullsize.png?alt=media">

\\

\therefore

Equation of line are

\begin{gathered}\frac{x-2}{1 / \sqrt{3}}=\frac{y+1}{1 / \sqrt{3}}=\frac{z-2}{1 / \sqrt{3}}\\ \Rightarrow \quad x-2=y+1=z-2=r \text{ (say) } \end{gathered}\therefore

Any point on the line is Q

\equiv(

r+2 r-1, r+2)

\because Q

lies on the plane

2 x+y+z=9

\begin{aligned}& \therefore \quad 2(r+2)+(r-1)+(r+2)=9 \\ & \Rightarrow 4 r+5=9 \Rightarrow r=1 \Rightarrow Q(3,0,3) \\ & \therefore P Q=\sqrt{(3-2)^2+(0+1)^2+(3-2)^2}\\ & =\sqrt{3} \end{aligned}

$$

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