Download App

Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry5 Marks
Question
Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-5}{4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{\text{y}-4}{1}=\frac{3-5}{3}$
✓

Answer

Given, equation of line is $\frac{\text{x}-5}{4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}=\lambda\text{ (say) }\dots(1)$ General point on line (1) is, $\big(4\lambda+5,4\lambda+7,-5\lambda-3\big)$ Another equation of line is, $\frac{\text{x}-8}{7}=\frac{\text{y}-4}{1}=\frac{3-5}{3}=\mu\text{ (say) }\dots(2)$ General point on line (2) is $\big(7\mu+8,\mu+4,3\mu+5\big)$ If line (1) and (2) intersecting, then there must have some common point to them, so, we must have value of $\lambda$ and $\mu$ such that $4\lambda+5=7\mu+8\Rightarrow4\lambda-7\mu=3\dots(3)$ $4\lambda+5=\mu+4\Rightarrow4\lambda-\mu=-3\dots(4)$ $-5\lambda-3=3\mu+5\Rightarrow-5\lambda-3\mu=8\dots(5)$ Solving equation (3) and (4) to find $\lambda$ and $\mu,$
$\mu=-1$ Put value of $\lambda$ in equation (3), $4\lambda-7\mu=3$ $4\lambda-7(-1)=3$ $4\lambda=3-7$ $\lambda=-1$ Put the value of $\lambda$ and $\mu$ in equation (5), $-5\lambda-3\mu=8$ $-5(-1)-3(-1)=8$ $5+3=8$ $\text{LHS = RHS}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1\text{ at }(\text{a}\sec\theta,\text{b}\tan\theta)$
→
Find:
$\int\frac{\text{dx}}{\text{x}^{3}(\text{x}^{5} + 1)^{3/5}}$
→
Find the general solution of the differential equation
$\frac{\text{dy}}{\text{dx}}-\text{y}=\sin\text{x}.$
→
Find the condition for the following set of curves to intersect orthogonally
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1\text{ and }\text{xy}=\text{c}^2$
→
Evaluate the following integrals:
$\int\frac{1}{1+\sqrt{\text{x}}}\text{dx}$
→
Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\sqrt{\text{y}^2-\text{x}^2}$
→
Calculate the area of the ragion bounded by $y^2= x$ and $x^2= y$.
→
A square piece of tin of side $18\ cm$ is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
→
If $\text{A} = \begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix},$ prove that $\text{A}'' = \begin{bmatrix}3^{n-1}&3^{n-1}&3^{n-1}\\3^{n-1}&3^{n-1}&3^{n-1}\\3^{n-1}&3^{n-1}&3^{n-1}\end{bmatrix}n \in \text{N}.$
→
If $\text{A}=\begin{bmatrix}3&1\\-1&2\end{bmatrix},$ show that $A^2 - 5A + 7I_2 = 0$.
→