Three-Dimensional Geometry — Class 12 Mathematics PYQs

If a line makes angles of $\frac{3\pi }{4}$, $\frac{\pi }{3}$ and $\theta$ with the positive directions of $x$, $y$ and $z$-axis respectively, then $\theta$ is

(A) $-\frac{\pi }{3}$ only
(B) $\frac{\pi }{3}$ only
(C) $\frac{\pi }{6}$
(D) $\pm \frac{\pi }{3}$

The equation of a line parallel to the vector $3\hat{i}+\hat{j}+2\hat{k}$ and passing through the point $(4,-3,7)$ is :

(A) $x=4t+3$, $y=-3t+1$, $z=7t+2$
(B) $x=3t+4$, $y=t+3$, $z=2t+7$
(C) $x=3t+4$, $y=t-3$, $z=2t+7$
(D) $x=3t+4$, $y=-t+3$, $z=2t+7$

The line $x=1+5\mu$, $y=-5+\mu$, $z=-6-3\mu$ passes through which of the following point ?

(A) $(1,-5,6)$
(B) $(1,5,6)$
(C) $(1,-5,-6)$
(D) $(-1,-5,6)$

(a) Verify that lines given by $\vec{r}=(1-\lambda )\hat{i}+(\lambda -2)\hat{j}+(3-2\lambda )\hat{k}$ and $\vec{r}=(\mu +1)\hat{i}+(2\mu -1)\hat{j}-(2\mu +1)\hat{k}$ are skew lines. Hence, find the shortest distance between the lines.

OR

(b) During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by $B=2\hat{i}+8\hat{j}$, $W=6\hat{i}+12\hat{j}$ and $F=12\hat{i}+18\hat{j}$ respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.

(a) Find the image $A^{\prime }$ of the point $A(1,6,3)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. Also, find the equation of the line joining $A$ and $A^{\prime }$.

OR

(b) Find a point $P$ on the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$ such that its distance from point $Q(2,4,-1)$ is 7 units. Also, find the equation of line joining $P$ and $Q$.

(a) Find the shortest distance between the lines :

$$\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\text{and}\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$$

OR

(b) Find the image $A^{\prime }$ of the point $A(2,1,2)$ in the line $l:\vec{r}=4\hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-\hat{j}-\hat{k})$. Also, find the equation of line joining $A{A}^{\prime }$. Find the foot of $\perp$ from point $A$ on the line $l$.

The coordinates of the foot of the perpendicular drawn from the point $(0,1,2)$ on the $x$-axis are given by :

If the direction cosines of a line are $\sqrt{3}k$, $\sqrt{3}k$, $\sqrt{3}k$, then the value of $k$ is :

Direction ratios of a vector parallel to line $\frac{x-1}{2}=\frac{-y}{1}=\frac{2z+1}{6}$ are :

The distance of point $P(a,b,c)$ from $y$-axis is :