Problem 49 - A Complete Resource Book in Mathematics for JEE Main

Question

Distance between two parallel planes \(2 x+y+2 z=8\) and \(4 x+2 y+4 z+5=0\) is (A) \(\frac{3}{2}\) (B) \(\frac{5}{2}\) (C) \(\frac{7}{2}\) (D) \(\frac{9}{2}\)

Step-by-Step Solution

Answer

The distance between the planes is \(\frac{7}{2}\), option (C).

Step 1: Identify the Normal Vector

The normal vector common to both planes can be identified by inspecting their coefficients, recognized as \((A, B, C)\). For the given planes, select \(2x + y + 2z = 8\) where the normal vector is \((2, 1, 2)\). Check the second equation \(4x + 2y + 4z + 5 = 0\) which simplifies to \(2x + y + 2z = -\frac{5}{2}\) confirming the normal vector is unchanged.

Step 2: Calculate Parallel Planes Distance Formula

Use the formula for the distance \(d\) between two parallel planes \(Ax + By + Cz = D_1\) and \(Ax + By + Cz = D_2\). The distance formula is: \[ d = \frac{|D_1 - D_2|}{\sqrt{A^2 + B^2 + C^2}} \] Where \(D_1 = 8\) and \(D_2 = -\frac{5}{2}\).

Step 3: Compute Values and Distance

Plug in the values into the distance formula: Substitute \(A = 2, B = 1, C = 2, D_1 = 8, D_2 = -\frac{5}{2}\): \[ d = \frac{|8 - (-\frac{5}{2})|}{\sqrt{2^2 + 1^2 + 2^2}} \] Simplify: \[ d = \frac{|8 + \frac{5}{2}|}{\sqrt{9}} = \frac{\frac{16}{2} + \frac{5}{2}}{3} = \frac{\frac{21}{2}}{3} = \frac{21}{6} = \frac{7}{2} \]

Step 4: Conclusion

The computed results show that the distance between the two parallel planes is \(\frac{7}{2}\). Refer to the provided options and confirm that \(\frac{7}{2}\) matches option (C).

Key Concepts

Normal Vector

When we talk about planes in three-dimensional space, the normal vector is a key concept to grasp. It is a vector that is perpendicular to the plane.
In mathematical expressions of planes, like in the equations we have here

• \(2x + y + 2z = 8\)
• \(4x + 2y + 4z + 5 = 0\)

the normal vector can be identified by looking at the coefficients of \(x\), \(y\), and \(z\). In simpler terms, the numbers in front of \(x, y, z\) are essentially the components of the normal vector.
For both planes, we notice that

• \(2\) with \(x\),
• \(1\) with \(y\),
• \(2\) with \(z\),

which forms the normal vector \((2, 1, 2)\). The fact that these components are the same in both equations is why the planes are parallel to each other. Understanding how to identify the normal vector is fundamental when dealing with planes and their relationships in space.

Parallel Plane Distance Formula

Distance between parallel planes can be calculated using a specific formula. This is where math becomes a nifty tool, giving us a method to determine exactly how far apart these planes are.
The formula for the distance \(d\) between two parallel planes is:\[d = \frac{|D_1 - D_2|}{\sqrt{A^2 + B^2 + C^2}}\] Here:

• \(D_1\) and \(D_2\) are the constants from the plane equations \(Ax + By + Cz = D_1\) and \(Ax + By + Cz = D_2\).
• \(A\), \(B\), and \(C\) come from the normal vector \((A, B, C)\) that we earlier identified.

This formula gives us a clean way to calculate the distance, emphasizing the neat correlation between geometry and algebra where vectors and distances interlink seamlessly.

Calculation of Distance

With the distance formula at hand, the next step is to dive into the actual calculation of distance using our known values. Start by substituting the values into the formula.
For instance, we have:

• \(D_1 = 8\) (from the first plane equation),
• \(D_2 = -\frac{5}{2}\) (adjusted from the second equation),
• Normal vector components \(A = 2\), \(B = 1\), and \(C = 2\).

Plug them into:\[d = \frac{|8 - (-\frac{5}{2})|}{\sqrt{2^2 + 1^2 + 2^2}}\]
Calculate each part step by step: first simplify the numerator:\[|8 + \frac{5}{2}| = \frac{21}{2}\]Then, compute the denominator:\[\sqrt{4 + 1 + 4} = \sqrt{9} = 3\] Finally, the distance is:\[d = \frac{\frac{21}{2}}{3} = \frac{21}{6} = \frac{7}{2}\] This shows the calculated distance between the two planes is \(\frac{7}{2}\), completing the process and confirming with option (C). This calculation shows how clear and systematic problem-solving methods help us find precise answers.