Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.
Solution and Explanation
The equation of the plane ZOX is y=0
Any plane parallel to it is of the form, y=a
Since, the y-intercept of the plane is 3,
∴ a=3
Thus, the equation of the required plane is y=3.
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
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- 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
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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 length of the diagonal of a rectangle with length 6 cm and breadth 8 cm.
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Questions Asked in CBSE CLASS XII exam
- If \( \int \frac{1}{2x^2} \, dx = k \cdot 2x + C \), then \( k \) is equal to:
- CBSE CLASS XII - 2025
- Integration
- A rectangular glass slab ABCD (refractive index 1.5) is surrounded by a transparent liquid (refractive index 1.25) as shown in the figure. A ray of light is incident on face AB at an angle \(i\) such that it is refracted out grazing the face AD. Find the value of angle \(i\).
- CBSE CLASS XII - 2025
- Refraction
- If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]
- Three students run on a racing track such that their speeds add up to 6 km/h. However, double the speed of the third runner added to the speed of the first results in 7 km/h. If thrice the speed of the first runner is added to the original speeds of the other two, the result is 12 km/h. Using the matrix method, find the original speed of each runner.
- Write a comprehensive report detailing the activities undertaken by students as part of the World Environment Day celebrations. You are Ranjana/Rajan, a student of class XII and a member of the school magazine editorial board. You may organise your report by using the following cues:
\includegraphics[width=0.75\linewidth]{6b.JPG- CBSE CLASS XII - 2025
- Report Writing
Concepts Used:
Plane
A surface comprising all the straight lines that join any two points lying on it is called a plane in geometry. A plane is defined through any of the following uniquely:
- Using three non-collinear points
- Using a point and a line not on that line
- Using two distinct intersecting lines
- Using two separate parallel lines
Properties of a Plane:
- In a three-dimensional space, if there are two different planes than they are either parallel to each other or intersecting in a line.
- A line could be parallel to a plane, intersects the plane at a single point or is existing in the plane.
- If there are two different lines that are perpendicular to the same plane then they must be parallel to each other.
- If there are two separate planes which are perpendicular to the same line then they must be parallel to each other.