Question

Find the image of the point (1, 6, 3) with respect to the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ is ___.

• A. $(1,3,5)$
• B. $(1,-6,11)$
• C. $(1,0,7)$
• D. $(1,2,3)$

Answer 1

The correct option is C $(1,0,7)$

Let P (1, 6, 3) be the given point and let L be the foot of the perpendicular from P to the given line.


The coordinates fo a general point on the given line are
$\frac{x-0}{1}=\frac{y-1}{2}=\frac{z-2}{3}i.e.x=\lambda ,y=2\lambda +1,z=3\lambda +2$
If the coordinates of L are$(\lambda ,2\lambda +1,3\lambda +2)$ then the direction ratios of PL are$\lambda -1,2\lambda -5,3\lambda -1.$
But the direction ratios of given line which is perpendicular to PL are $1,2,3.Therefore(\lambda -1)1+(2\lambda -5)2+(3\lambda -1)3=0,$ which gives $\lambda =1.$ Hence coordinates of L are $(1,3,5).$
Let $Q(x_1,y_1,z_1)$ be the image fo P (1, 6, 3) in the given line then L is the md -point of PQ. Therefore,
$\frac{x_1+1}{2}=1,\frac{y_1+6}{2}=3,\frac{z_1+3}{2}=5\Rightarrow x_1=1,y_1=0,z_1=7$