Three Dimensional Geometry

Given :
Equation of the plane :


Standard form of Equation of plane :

Therefore, x + y + z = 1 ……. (i)
Coordinates of Q and S :
Q = (10, 15, 20)
S = (α, β, γ)
∴

∴ α = 10 = β = -15 γ - 20 =
Therefore, the values are as follows :

∴ 3 (α + β) = −101 so, option (A) is correct.
3(β + γ) =−71 so, option (B) is correct.
3(γ + α) = −86 so, option (C) is correct.
3(α+β+γ)=−129 so, option (D) is incorrect.

So, the correct options are (A), (B) and (C).

Step 1: Understanding Reflection and General Formula

The formula for the reflection of a point with respect to a plane is given by: In our case, we need to find the coordinates of based on the given reflection formula. To start, we need to express the equation of the plane in standard form.

Step 2: Equation of the Plane

The equation of the plane is given by: The equation of the plane in standard form can be written as: The general form of the plane equation is: We now compute the reflection coordinates using this plane equation and the reflection formula.

Step 3: Calculating the Reflected Position Vector

The position vectors of and are provided: Using the formula for the reflection and substituting the known values, we compute:

Step 4: Checking the Given Options

Now we compare the results with the given options:

Option A:

This is correct, as it matches the result from the reflection calculation.

Option B:

This is also correct.

Option C:

This is correct as well.

Option D:

This does not match the reflection calculations, so it is not correct.

Final Answer:

The correct options are: A, B, and C.

Step 1: Understanding the Planes

The equation of a plane is given by:

, where are the coefficients of the normal vector to the plane.

For plane :

We have the equation . The normal vector to this plane is:

.

For plane :

We have the equation . The normal vector to this plane is:

.

Step 2: Finding the Line of Intersection

The line of intersection of two planes is given by the cross product of their normal vectors.

Let’s find the cross product of and :

,

We compute the cross product as follows:

Expanding the determinant:

Thus, the direction vector of the line of intersection is:

.

Step 3: Equation of the Line

The line of intersection can be written in parametric form as: where is a point on the line, and is the direction vector. Here, the direction vector is . The parametric equations of the line become: This gives the line equations as: where is a parameter. This corresponds to the straight line that could be an edge of the tetrahedron formed by the intersection of the two planes.

Step 4: Matching with Given Options

From the given options, we check the direction vectors and equations:

Option A:

This matches the form we derived, and so it is correct.

Option B:

This doesn't match the derived equations, so it’s not correct.

Option C:

This matches the form, and so it is correct.

Option D:

This matches the form, and so it is correct.

Final Answer:

The correct options are: A, C, D.

1. Equation of Line Parallel to Through Point :
We are given the equation of a line and the point . The equation of the line parallel to the given one can be written as:

(Let)

Substituting in the above equation, we get:

2. Equation of Line Through Perpendicular to :
The equation of the line through perpendicular to is given by:

(Let)

3. Putting Any Point in :
We substitute the coordinates of the point in and simplify the calculations.

4. Point :
After simplification, we find that point lies on . Now, let's calculate the distances:

5. Distance Between Points and :
We use the distance formula to find :

6. Distance Between Points and :
Similarly, we calculate :

7. Distance Between Points and :
Next, we calculate :

8. Final Calculation:
The total distance is the sum of , , and :

Final Answer:

• (A)
• (B)
• (C) Centroid:
• (D)

For the given problem, we have two lines and . These lines intersect at a point . We need to determine the matched items from List-I to List-II based on the conditions provided.

1. Finding Intersection: Set the parameter for as , , and parameter for as , , . Equate to satisfy all three coordinates simultaneously for intersection.
2. Solving Equations: Solve and equations:
3. Calculate :
   • From ,
   • From , substitute :
4. Match Lists: Evaluate , unit normal , and calculate corresponding values:
   • Unit normal possibilities:
     • Cross product of direction vectors and gives a normal vector
     • Normalize
   • Calculate :

Thus, the correct matching is (P) → (3), (Q) → (4), (R) → (1), (S) → (5).