JEE Mathematics 2023 Question with Solution

Given: The plane $P$ passes through $(5, 3, 0)$, $(13, 3, -2)$, and $(1, 6, 2)$. The distances of $A(3, 4, \alpha)$ and $B(2, \alpha, a)$ from the plane are $2$ and $3$ respectively.

Find: The positive value of $a$.

From the solution, the normal vector is obtained using the determinant

$$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 8 & 0 & -2 \\ 4 & -3 & -2 \end{vmatrix} = \hat{i}(-6) + 8\hat{j} - 24\hat{k}$$

So the normal vector of the plane is taken as

$$3\hat{i} - 4\hat{j} + 12\hat{k}$$

Hence the equation of the plane is

$$3x - 4y + 12z = 3$$

Now for the point $A(3,4,\alpha)$, the distance formula gives

$$\frac{|3(3) - 4(4) + 12\alpha - 3|}{\sqrt{3^2 + (-4)^2 + 12^2}} = 2$$

Since

$$|9 - 16 + 12\alpha - 3| = |12\alpha - 10|$$

and

$$\sqrt{3^2 + (-4)^2 + 12^2} = \sqrt{9 + 16 + 144} = 13$$

we get

$$\frac{|12\alpha - 10|}{13} = 2$$

Thus

$$|12\alpha - 10| = 26$$

So

$$12\alpha - 10 = 26 \Rightarrow \alpha = 3$$

or

$$12\alpha - 10 = -26 \Rightarrow \alpha = -\frac{4}{3}$$

Since $\alpha \in \mathbb{N}$, we take

$$\alpha = 3$$

Now substitute $\alpha = 3$ in $B(2,\alpha,a)$, so the point becomes $B(2,3,a)$. Its distance from the plane is $3$, hence

$$\frac{|3(2) - 4(3) + 12a - 3|}{13} = 3$$

That is,

$$|6 - 12 + 12a - 3| = |12a - 9|$$

So

$$\frac{|12a - 9|}{13} = 3$$

and therefore

$$|12a - 9| = 39$$

This gives

$$12a - 9 = 39 \Rightarrow a = 4$$

or

$$12a - 9 = -39 \Rightarrow a = -\frac{5}{2}$$

The positive value is $4$.

Therefore, the value obtained from the working is $4$. This corresponds to option C, although the solution incorrectly labels the correct option as D.

Using the plane equation

$$3x - 4y + 12z - 3 = 0$$

the perpendicular distance of a point $(x_1,y_1,z_1)$ from the plane is

$$\frac{|3x_1 - 4y_1 + 12z_1 - 3|}{\sqrt{3^2 + (-4)^2 + 12^2}}$$

For $A(3,4,\alpha)$,

$$\frac{|9 - 16 + 12\alpha - 3|}{13} = 2$$

which reduces to

$$|12\alpha - 10| = 26$$

Since $\alpha$ is a natural number, $\alpha = 3$.

Then for $B(2,\alpha,a) = B(2,3,a)$,

$$\frac{|6 - 12 + 12a - 3|}{13} = 3$$

so

$$|12a - 9| = 39$$

Hence

$$a = 4 \quad \text{or} \quad a = -\frac{5}{2}$$

The positive value is $4$, so the correct option is C.