Equation of a line in space — practice questions

Question 1 · difficulty L2 · understanding

$\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(a \hat{i}-\hat{j}), a \neq 0$ and $\vec{r}=(4 \hat{i}-\hat{k})+\mu(2 \hat{i}+a \hat{k})$ from the origin is :

• A. 5
• B. 10
• C. 17
• D. 26

Question 2 · difficulty L2 · understanding

The shortest distance between the lines $$\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$$ and $$\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$$ is

• A. $$8 \sqrt{3}$$
• B. $$6 \sqrt{3}$$
• C. $$5 \sqrt{3}$$
• D. $$4 \sqrt{3}$$

Question 3 · difficulty L3 · understanding

Let the foot of perpendicular from the point $(\lambda, 2,3)$ on the line $\frac{x-4}{1}=\frac{y-9}{2}=\frac{z-5}{1}$ be the point ( $1, \mu, 2$ ). Then the distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$ and $\frac{x-\lambda}{2}=\frac{y-\mu}{3}=\frac{z+5}{6}$ is equal to :

• A. $\frac{12}{7}$
• B. $\frac{\sqrt{145}}{7}$
• C. $$\frac{\sqrt{146}}{7}$$
• D. $$\frac{\sqrt{143}}{7}$$

Question 4 · difficulty L3 · understanding

Let the image of the point $\mathrm{P}(1,6, a)$ in the line $\mathrm{L}: \frac{x}{1}=\frac{y-1}{2}=\frac{z-a+1}{b}, b>0$, be $\left(\frac{a}{3}, 0, a+c\right)$. If $\mathrm{S}(\alpha, \beta, \gamma), \alpha>0$, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is $2 \sqrt{14}$, then $\alpha+\beta+\gamma$ is equal to:

• A. 19
• B. 20
• C. 21
• D. 22

Question 5 · difficulty L3 · understanding

Let a line L be perpendicular to both the lines $\mathrm{L}_1: \frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ and $\mathrm{L}_2: \frac{x-2}{1}=\frac{y-4}{4}=\frac{z-6}{7}$. If $\theta$ is the acute angle between the lines L and $\mathrm{L}_3: \frac{x-\frac{8}{7}}{2}=\frac{y-\frac{4}{7}}{1}=\frac{z}{2}$, then $\tan \theta$ is equal to:

• A. $\frac{3}{2} \sqrt{2}$
• B. $\frac{5}{2} \sqrt{2}$
• C. $\frac{5}{3} \sqrt{2}$
• D. $\frac{4}{3} \sqrt{2}$

Question 6 · difficulty L3 · understanding

Let a triangle PQR be such that P and Q lie on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ and are at a distance of 6 units from $R(1,2,3)$. If $(\alpha, \beta, \gamma)$ is the centroid of $\Delta P Q R$, then $\alpha+\beta+\gamma$ is equal to :

• A. 4
• B. 5
• C. 6
• D. 8

Question 7 · difficulty L3 · understanding

If the distance of the point $(a, 2,5)$ from the image of the point $(1,2,7)$ in the line $\frac{x}{1}=\frac{y-1}{1}=\frac{z-2}{2}$ is 4 , then the sum of all possible values of $a$ is equal to :

• A. 11
• B. 9
• C. 6
• D. 4

Question 8 · difficulty L3 · understanding

The square of the distance of the point $\mathrm{P}(5,6,7)$ from the line $\frac{x-2}{2}=\frac{y-5}{3}=\frac{z-2}{4}$ is equal to:

• A. 3
• B. 5
• C. 6
• D. 8

Question 9 · difficulty L3 · understanding

The shortest distance between the lines $$\vec{r}=\left(\frac{1}{3} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\frac{8}{3} \hat{\mathrm{k}}\right)+\lambda(2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$$ and $\vec{r}=\left(-\frac{2}{3} \hat{\mathrm{i}}-\frac{1}{3} \hat{\mathrm{k}}\right)+\mu(\hat{\mathrm{j}}-\hat{\mathrm{k}}), \lambda, \mu \in \mathbb{R}$, is:

• A. $\sqrt{5}$
• B. 3
• C. $2 \sqrt{3}$
• D. $\sqrt{15}$

Question 10 · difficulty L3 · understanding

If $\left(2 \alpha+1, \alpha^2-3 \alpha, \frac{\alpha-1}{2}\right)$ is the image of $(\alpha, 2 \alpha, 1)$ in the line $\frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1}$, then the possible value(s) of $\alpha$ is (are)

• A. Only 3
• B. Only 3 and - 1
• C. Only $3, \frac{1}{4}$ and -1
• D. Only 3 and $\frac{1}{4}$