Find the equation of the plane through the intersection of the planes
3x-y+2z-4 =0 and x+y+z-2=0 and the point (2, 2, 1).
Find the equation of the plane through the intersection of the planes
3x-y+2z-4 =0 and x+y+z-2=0 and the point (2, 2, 1).
Solution and Explanation
The equation of any plane through the intersection of the planes,
3x-y+2z-4=0 and x+y+z-2=0, is
(3x-y+2z-4)+ \(\alpha\) (x+y+z-2)=0, where \(\alpha \in\) R...(1)
The plane passes through the point (2 ,2,1).
Therefore, this point will satisfy equation(1).
∴ (3×2-2+2×1-4)+α(2+2+1-2)=0
\(\Rightarrow \) 2+3\(\alpha\) =0
\(\Rightarrow \alpha=-\frac{2}{3}\)
Substituting \(\alpha=-\frac{2}{3}\) in equation(1), we obtain
(3x-y+2z-4)-\(\frac{2}{3}\) (x+y+z-2)=0
\(\Rightarrow 3\) (3x-y+2z-4)-2(x+y+z-2)=0
\(\Rightarrow\) (9x-3y+6z-12)-2(x+y+z-2)=0
\(\Rightarrow \) 7x-5y+4z-8=0
This is the required equation of the plane.
Top Questions on Three Dimensional Geometry
- If the distances of the point \( (1,2,a) \) from the line \[ \frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} \] along the lines \[ L_1:\ \frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b} \quad \text{and} \quad L_2:\ \frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c} \] are equal, then \( a+b+c \) is equal to:
- JEE Main - 2026
- Mathematics
- Three Dimensional Geometry
- The value of the integral \( \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}} \) is :
- JEE Main - 2026
- Mathematics
- Three Dimensional Geometry
Let the lines $L_1 : \vec r = \hat i + 2\hat j + 3\hat k + \lambda(2\hat i + 3\hat j + 4\hat k)$, $\lambda \in \mathbb{R}$ and $L_2 : \vec r = (4\hat i + \hat j) + \mu(5\hat i + + 2\hat j + \hat k)$, $\mu \in \mathbb{R}$ intersect at the point $R$. Let $P$ and $Q$ be the points lying on lines $L_1$ and $L_2$, respectively, such that $|PR|=\sqrt{29}$ and $|PQ|=\sqrt{\frac{47}{3}}$. If the point $P$ lies in the first octant, then $27(QR)^2$ is equal to}
- JEE Main - 2026
- Mathematics
- Three Dimensional Geometry
- Let a line $L$ passing through the point $P(1,1,1)$ be perpendicular to the lines \[ \frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1} \quad \text{and} \quad \frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}. \] Let the line $L$ intersect the $yz$-plane at the point $Q$.
Another line parallel to $L$ and passing through the point $S(1,0,-1)$ intersects the $yz$-plane at the point $R$.
Then the square of the area of the parallelogram $PQRS$ is equal to
- JEE Main - 2026
- Mathematics
- Three Dimensional Geometry
- Let \( L \) be the line \[ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z+3}{6} \] and let \( S \) be the set of all points \( (a,b,c) \) on \( L \), whose distance from the line \[ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z-9}{0} \] along the line \( L \) is \( 7 \). Then \[ \sum_{(a,b,c)\in S} (a+b+c) \] is equal to
- JEE Main - 2026
- Mathematics
- Three Dimensional Geometry
Questions Asked in CBSE CLASS XII exam
- 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
- Considering the case of magnetic field produced by air-filled current carrying solenoid, show that the magnetic energy density of a magnetic field \( B \) is \( \frac{B^2}{2\mu_0} \).
- CBSE CLASS XII - 2025
- Magnetic Field
- The slope of the curve \( y = -x^3 + 3x^2 + 8x - 20 \) is maximum at:
- CBSE CLASS XII - 2025
- Maxima and Minima
- Three batteries E1, E2, and E3 of emfs and internal resistances (4 V, 2 \(\Omega\)), (2 V, 4 \(\Omega\)) and (6 V, 2 \(\Omega\)) respectively are connected as shown in the figure. Find the values of the currents passing through batteries E1, E2, and E3.
- CBSE CLASS XII - 2025
- Kirchhoff's Laws
- Two wires made of the same material have the same length \( l \) but different cross-sectional areas \( A_1 \) and \( A_2 \). They are connected together with a cell of voltage \( V \). Find the ratio of the drift velocities of free electrons in the two wires when they are joined in: (i) series, and (ii) parallel.
- CBSE CLASS XII - 2025
- Current electricity
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.