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  • If the image of the point P(1, −2, 3) in the plane, 2x+3y−4z+22=0 measured parallel to the line, <img alt="\frac{x}{1}= \frac{y}{4}= \frac{z}{5}" src="https://learn.careers360.com/latex-image/?%5Cdpi%7B100%7D%20%5Cfrac%7Bx%7D%7B1%7D%3D%20%5

If the image of the point P(1, −2, 3) in the plane, 2x+3y−4z+22=0 measured parallel to the line, \frac{x}{1}= \frac{y}{4}= \frac{z}{5}    is Q, then PQ is equal to :
Option: 1 2\sqrt{42}

Option: 5 \sqrt{42}

Option: 9 6\sqrt{5}

Option: 13 3\sqrt{5}
 

Answers (1)

best_answer

 

Intersection of line and plane -

Let the line

\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c} plane

a_{1}x+b_{1}y+c_{1}z+d=0 intersect at P

to find P assume general point on line as \left ( x_{1}+\lambda a_{1}y_{1} +\lambda b_{1}z_{1}+\lambda c_{1}\right )

now put it in plane to find \lambda,

a_{1}\left ( x_{1}+\lambda a \right )+b_{1}\left ( y_{1}+\lambda b \right )+c_{1}\left ( z_{1}+\lambda c \right )+d=0

-

 

 

Distance formula -

The distance between two points A(x_{1},y_{1},z_{1})\, and \, B(x_{2},y_{2},z_{2}) is =\sqrt{\left ( x_{2}-x_{1} \right )^{2}+{\left ( y_{2}-y_{1} \right )^{2}}+{\left ( z_{2}-z_{1} \right )^{2}}}

 

- wherein

\vec{OA}= x_{1}\hat{i}+y_{1}\hat{j}+z_{1}\hat{k}

\vec{OB}= x_{2}\hat{i}+y_{2}\hat{j}+z_{2}\hat{k}

\underset{AB}{\rightarrow}    = \underset{OB}{\rightarrow} - \underset{AB}{\rightarrow}

\underset{AB}{\rightarrow}  = \left ( x_{2}-x_{1} \right )\hat{i}+\left ( y_{2}-y_{1} \right )\hat{j}+\left ( z_{2}-z_{1} \right )\hat{k}

AB= \left | \underset{AB}{\rightarrow} \right |

AB= \sqrt{\left ( x_{2}-x_{1} \right )^{2}+\left ( y_{2}-y_{1} \right )^{2}+\left ( z_{2}-z_{1} \right )^{2}}

 

 

 

 

PQ=2PM

Now, for finding M

\frac{x}{1}=\frac{y}{4}=\frac{z}{5}=k(say)

\Rightarrow x=k,y=4k,z=5k

Putting in equation of plane,

k=\frac{11}{3}

So, M=\left (\frac{11}{3} ,\frac{44}{3},\frac{55}{3} \right )

\therefore PM=\sqrt{\frac{64}{9}+\frac{2500}{9}+\frac{2116}{9}}= 2\sqrt{130}

\Rightarrow PQ=4\sqrt{130}

 

 

 

 

 

Posted by

vishal kumar

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