CBSE 2023 · Region 1 · Set 1 · Q34 · 5 marks
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.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 $\displaystyle \mathrm{B}$.
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.
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}}$
Three Dimensional GeometryShortest Distance between Two LinesApplylong_answermedium
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…2025 · asked 3×
- 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 · asked 3×
- If a line makes an angle of (π)/4 with the positive directions of both x -axis and z -axis, then the angle…2024 · asked 3×
- Assertion: If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin^2 α+ sin^2…2023 · asked 3×
- 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…2026 · asked 3×
- Find the equation of the plane passing through points (2, 1, 0), (3, -2, -2) and (1, 1, -7). Also, obtain its…2022 · asked 3×
- Direction ratios of a vector parallel to line x-1/2=-y=(2 z+1)/6 are:2024 · asked 3×
- Assertion ( A ): The lines r= a 1+λ b 1 and r= a 2+μ b 2 are perpendicular, when b 1 · b 2=0. Reason (R): The…2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.