✓ Board-verified
Mathematics · 2023 · 5 marks
CBSE 2023 · Region 1 · Set 1 · Q34
(a) Find the vector and the Cartesian equations of a line passing through the point ( 1,2,-4 ) and parallel to the line joining the points A (3,3,-5)…
(a)
Find the vector and the Cartesian equations of a line passing through the point ( $\displaystyle 1,2,-4$ ) and parallel to the line joining the points $\displaystyle \mathrm{A}(3,3,-5)$ and $\displaystyle \mathrm{B}(1,0,-11)$. Hence, find the distance between the two lines.
OR
(b)
Find the equations of the line passing through the points $\displaystyle \mathrm{A}(1,2,3)$ and $\displaystyle \mathrm{B}(3,5,9)$. Hence, find the coordinates of the points on this line which are at a distance of $\displaystyle 14$ units from point $\displaystyle \mathrm{B}$.
Marking-scheme solution
(a)
Vector equation of required line through ( $\displaystyle 1,2,-4$ ) is
$$\vec{r}=\hat{\mathrm{i}}+$\displaystyle 2$ \hat{\mathrm{j}}-$\displaystyle 4$ \hat{\mathrm{k}}+\lambda($\displaystyle 2$ \hat{\mathrm{i}}+$\displaystyle 3$ \hat{\mathrm{j}}+$\displaystyle 6$ \hat{\mathrm{k}})
$$and cartesian equation: $\displaystyle \frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$
Equation of line through $\displaystyle \mathrm{A}(3,3,-5)$ and $\displaystyle \mathrm{B}(1,0,-11)$ is
$$\vec{r}=$\displaystyle 3$ \hat{\mathrm{i}}+$\displaystyle 3$ \hat{\mathrm{j}}-$\displaystyle 5$ \hat{\mathrm{k}}+\mu($\displaystyle 2$ \hat{\mathrm{i}}+$\displaystyle 3$ \hat{\mathrm{j}}+$\displaystyle 6$ \hat{\mathrm{k}})
$$Distance between parallel lines is given by $\displaystyle \mathrm{d}=\frac{\left|\left(\overrightarrow{a_{2}}-\overrightarrow{a_{1}}\right) \times \vec{b}\right|}{|\vec{b}|}$
Here $\displaystyle \vec{b}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}, \vec{a}_{1}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}, \overrightarrow{a_{2}}=3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}$
$$\left(\overrightarrow{a_{$\displaystyle 2$}}-\vec{a}_{$\displaystyle 1$}\right)=$\displaystyle 2$ \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}
$$$\left(\overrightarrow{a_{2}}-\overrightarrow{a_{1}}\right) \times \vec{b}=9 \hat{\mathrm{i}}-14 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$
All Three Dimensional Geometry questions2023 question paperNCERT solutions: Three Dimensional GeometryPractice Three Dimensional Geometry →
More from Three Dimensional Geometry
- Find the image A^′ of the point A(1,6,3) in the line x/1=(y-1)/2=(z-2)/3. Also, find the equation of the line joining A and A^′. OR Find a point P on…2025
- Find the values of λ, for which the distance of point (2,1, λ) from plane 3 x+5 y+ 4 z=11 is 2 √2 units.2022
- Find the value of λ if the following lines are perpendicular to each other: l_1: (1-x)/-3=(3 y-2)/(2 λ)=(z-3)/3; l_2: (x-1)/(3 λ)=(1-y)/1=(2 z-5)/32025
- The value of λ for which the angle between the lines r= i+ j+ k+p(2 i+ j+2 k) and r=(1+q) i+(1+q λ) j+(1+q) k is (π)/2 is:2023
- If a line makes an angle of (π)/4 with the positive directions of both x -axis and z -axis, then the angle which it makes with the positive direction…2024
- Assertion: If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin^2 α+ sin^2 β+ sin^2 γ=2. Reason (R): The sum of…2023
- If the lines (x-3)/1=(1-y)/1=(z+2)/p and (2-x)/3=(y+1)/5=(z+56)/(2 p) are perpendicular to each other, then find the value(s) of p. OR Find the…2026
- Find the equation of the plane passing through points (2, 1, 0), (3, -2, -2) and (1, 1, -7). Also, obtain its distance from the origin. OR Find the…2022
CBSE Class 12 Mathematics past-paper question from the 2023 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.