Question

Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes $x - y - z - 1 = 0$ and $2x + y - 3z + 4 = 0$, is :

• A. $3x - 4z + 3 = 0$
• B. $-x + 2y + 2z - 3 = 0$
• C. $3x - y - 5z + 2 = 0$
• D. $4x - y - 5z + 2 = 0$

Hint:

When dealing with a family of planes $P_1 + \lambda P_2 = 0$, if the final equation looks complicated, try verifying the options by checking their distance from the origin and whether they contain a point on the line of intersection (found by setting one coordinate to zero).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The equation of a plane passing through the intersection of two planes $P_1 = 0$ and $P_2 = 0$ is given by $P_1 + \lambda P_2 = 0$. We then use the distance formula from the origin to find $\lambda$.
Step 2: Key Formula or Approach:
Perpendicular distance of plane $ax + by + cz + d = 0$ from $(0,0,0)$ is $\frac{|d|}{\sqrt{a^2 + b^2 + c^2}}$.
Step 3: Detailed Explanation:
The equation of the required plane is:
\[ (x - y - z - 1) + \lambda(2x + y - 3z + 4) = 0 \]
\[ (1 + 2\lambda)x + (\lambda - 1)y + (-1 - 3\lambda)z + (4\lambda - 1) = 0 \]
Distance from origin $(0,0,0)$ is $\sqrt{\frac{2}{21}}$:
\[ \frac{|4\lambda - 1|}{\sqrt{(1+2\lambda)^2 + (\lambda-1)^2 + (-1-3\lambda)^2}} = \sqrt{\frac{2}{21}} \]
Squaring both sides:
\[ \frac{16\lambda^2 - 8\lambda + 1}{1 + 4\lambda^2 + 4\lambda + \lambda^2 - 2\lambda + 1 + 1 + 9\lambda^2 + 6\lambda} = \frac{2}{21} \]
\[ \frac{16\lambda^2 - 8\lambda + 1}{14\lambda^2 + 8\lambda + 3} = \frac{2}{21} \]
\[ 336\lambda^2 - 168\lambda + 21 = 28\lambda^2 + 16\lambda + 6 \]
\[ 308\lambda^2 - 184\lambda + 15 = 0 \]
Solving for $\lambda$ using the quadratic formula:
\[ \lambda = \frac{184 \pm \sqrt{184^2 - 4(308)(15)}}{2(308)} = \frac{184 \pm \sqrt{33856 - 18480}}{616} = \frac{184 \pm 124}{616} \]
This gives $\lambda = \frac{308}{616} = \frac{1}{2}$ or $\lambda = \frac{60}{616} = \frac{15}{154}$.
Using $\lambda = \frac{1}{2}$ in the plane equation:
\[ (1 + 2(1/2))x + (1/2 - 1)y + (-1 - 3/2)z + (4(1/2) - 1) = 0 \]
\[ 2x - \frac{1}{2}y - \frac{5}{2}z + 1 = 0 \]
Multiplying by 2:
\[ 4x - y - 5z + 2 = 0 \]
Step 4: Final Answer:
The equation of the plane is $4x - y - 5z + 2 = 0$.