NIMCET 2007 — Mathematics PYQ

To find the perpendicular distance between two parallel planes, we must first express both plane equations in standard format where the coefficients of $x$, $y$, and $z$ are identical.

The general formula for the distance $d$ between two parallel planes $Ax+By+Cz+D_1=0$ and $Ax+By+Cz+D_2=0$ is:

$$d=\frac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}$$

Step 1: Standardize the Equations

Let's look at our given equations:

• Plane 1: $2x+y+2z=8⟹2x+y+2z-8=0$

• Plane 2: $4x+2y+4z+5=0$

Notice that the coefficients of Plane 2 are twice the coefficients of Plane 1. We can divide the entire equation of Plane 2 by $2$ to match the coefficients:

$$\frac{4x+2y+4z+5}{2}=\frac{0}{2}$$

$$2x+y+2z+\frac{5}{2}=0$$

Step 2: Identify the Values

Now both equations have identical $Ax+By+Cz$ components:

• $A=2$

• $B=1$

• $C=2$

• $D_1=-8$

• $D_2=\frac{5}{2}$

Step 3: Apply the Distance Formula

Substitute these values into the parallel planes distance formula:

$$d=\frac{\left|-8-\frac{5}{2}\right|}{\sqrt{2^2+1^2+2^2}}$$

Simplify the numerator:

$$\left|-8-\frac{5}{2}\right|=\left|\frac{-16-5}{2}\right|=\left|-\frac{21}{2}\right|=\frac{21}{2}$$

Simplify the denominator:

$$\sqrt{4+1+4}=\sqrt{9}=3$$

Step 4: Calculate the Final Value

Combine the simplified parts:

$$d=\frac{\frac{21}{2}}{3}$$

$$d=\frac{21}{2\times 3}=\frac{7}{2}$$

Final Answer

The correct option is (a) 7/2.