Question

The image of the line $\frac{x}{2}=\frac{y - 1}{5}=\frac{z + 1}{3}$ in the plane $x+y+2z=3$ meets the $xz-$ plane at the point $\left(a , b , c\right),$ then the value of $c$ is equal to

• A. $\frac{11}{6}$
• B. $\frac{129}{6}$
• C. $\frac{115}{6}$
• D. $\frac{232}{3}$

Answer 1

Answer: (B)

Any general point on the line is $\left(2 \lambda , 5 \lambda + 1,3 \lambda - 1\right)$

On satisfying this point on the plane, we get,
$2\lambda +5\lambda +1+6\lambda -2=3$
$13\lambda =4\Rightarrow \lambda =\frac{4}{13}$
So, coordinates of the point $A^{'}$ are $\left(\frac{8}{13} , \frac{33}{13} , \frac{- 1}{13}\right)$
This point also lies on the image of the line
Image of point $\left(0,1 , - 1\right)$ also lies on the image of the line
$\frac{x - 0}{1}=\frac{y - 1}{1}=\frac{z + 1}{2}=-2\frac{\left(- 4\right)}{6}$
$x=\frac{4}{3},y=\frac{7}{3},z=\frac{5}{3}$
Point is $\left(\frac{4}{3} , \frac{7}{3} , \frac{5}{3}\right)$
Equation of image of the line is
$\frac{x - \frac{4}{3}}{28}=\frac{y - \frac{7}{3}}{- 8}=\frac{z - \frac{5}{3}}{68}$
For $xz-$ plane, putting $y=0$ , we get,
$\frac{x - \frac{4}{3}}{28}=\frac{7}{24}=\frac{z - \frac{5}{3}}{68}$
$\Rightarrow z=\frac{129}{6}$