The point $A(3, 1, 6)$ is the mirror image of the point $B(1, 3, 4)$ in the plane $x - y + z = 5$.
The plane $x - y + z = 5$ bisects the line segment joining $A(3, 1, 6)$ and $B(1, 3, 4)$.
- Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1
- Statement-1 is true, Statement-2 is false
- Statement-1 is false, Statement-2 is true
- Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1
The Correct Option is A
Solution and Explanation
Top Questions on Three Dimensional Geometry
Show that the following lines intersect. Also, find their point of intersection:
Line 1: \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \]
Line 2: \[ \frac{x - 4}{5} = \frac{y - 1}{2} = z \]
- CBSE CLASS XII - 2025
- CBSE Compartment XII - 2025
- Mathematics
- Three Dimensional Geometry
The vector equations of two lines are given as:
Line 1: \[ \vec{r}_1 = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(4\hat{i} + 6\hat{j} + 12\hat{k}) \]
Line 2: \[ \vec{r}_2 = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(6\hat{i} + 9\hat{j} + 18\hat{k}) \]
Determine whether the lines are parallel, intersecting, skew, or coincident. If they are not coincident, find the shortest distance between them.
- CBSE CLASS XII - 2025
- CBSE Compartment XII - 2025
- Mathematics
- Three Dimensional Geometry
Determine the vector equation of the line that passes through the point \( (1, 2, -3) \) and is perpendicular to both of the following lines:
\[ \frac{x - 8}{3} = \frac{y + 16}{7} = \frac{z - 10}{-16} \quad \text{and} \quad \frac{x - 15}{3} = \frac{y - 29}{-8} = \frac{z - 5}{-5} \]
- CBSE CLASS XII - 2025
- CBSE Compartment XII - 2025
- Mathematics
- Three Dimensional Geometry
- Find the value of λ, if the points A(−1,−1,2), B(2,8,λ), C(3,11,6) are collinear.
- CBSE CLASS XII - 2025
- CBSE Compartment XII - 2025
- Mathematics
- Three Dimensional Geometry
- Find the length of the diagonal of a rectangle with length 6 cm and breadth 8 cm.
- MHT CET - 2025
- Mathematics
- Three Dimensional Geometry
Questions Asked in AIEEE exam
- The ratio of number of oxygen atoms (O) in 16.0 g ozone $(O_3), \,28.0\, g$ carbon monoxide $(CO)$ and $16.0$ oxygen $(O_2)$ is (Atomic mass: $C = 12,0 = 16$ and Avogadro?? constant $N_A = 6.0 x 10^{23}\, mol^{-1}$)
- AIEEE - 2012
- Mole concept and Molar Masses
- This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. It is not possible to make a sphere of capacity $1$ farad using a conducting material. It is possible for earth as its radius is $6.4 \times 10^6\,m$.
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- electrostatic potential and capacitance
- A steel wire can sustain $100\, kg$ weight without breaking. If the wire is cut into two equal parts, each part can sustain a weight of
- AIEEE - 2012
- mechanical properties of solids
- The limiting line in Balmer series will have a frequency of (Rydberg constant, $R_{\infty}=3.29\times10^{15}$ cycles/s)
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- Which of the plots shown in the figure represents speed (v) of the electron in a hydrogen atom as a function of the principal quantum number (n)?
Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
