Question

Consider a line $L$ passing through the points $P(1, 2, 1)$ and $Q(2, 1, -1)$. If the mirror image of the point $A(2, 2, 2)$ in the line $L$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta + 6\gamma$ is equal to .

Answer 1

Step 1: Direction Ratios of Line $L$:

The direction ratios of line $L$ passing through $P(1,2,1)$ and $Q(2,1,-1)$ are: $\text{DR's of } L = (2 - 1, 1 - 2, -1 - 1) = 1 : -1 : -2.$

Step 2: Direction Ratios of $AB$:

Let $B(\alpha, \beta, \gamma)$ be the mirror image of $A(2,2,2)$ in line $L$. Then, the direction ratios of $AB$ are: $(\alpha - 2, \beta - 2, \gamma - 2).$

Step 3: Perpendicular Condition:

Since $AB$ is perpendicular to line $L$, we have: $1(\alpha - 2) - 1(\beta - 2) - 2(\gamma - 2) = 0.$ Simplifying, we get: \alpha - \beta - 2\gamma = -4. \tag{1}

Step 4: Finding the Midpoint $C$ of $AB$:

The midpoint $C$ of $AB$ lies on the line $L$, so: $C = \left(\frac{\alpha + 2}{2}, \frac{\beta + 2}{2}, \frac{\gamma + 2}{2}\right).$

Step 5: Direction Ratios of $PC$ and Parallel Condition:

The direction ratios of $PC$ are: $\left(\frac{\alpha - 2}{2}, \frac{\beta - 2}{2}, \frac{\gamma - 2}{2}\right).$ Since line $L$ is parallel to $PC$, we have: $\frac{\alpha - 2}{2} : \frac{\beta - 2}{2} : \frac{\gamma - 2}{2} = 1 : -1 : -2.$ This gives: $\alpha - 2 = -2K, \quad \beta - 2 = 2K, \quad \gamma - 2 = 4K.$ Solving for $\alpha$, $\beta$, and $\gamma$, we get: $\alpha = -2K + 2, \quad \beta = 2K + 2, \quad \gamma = 4K + 2.$

Step 6: Substitute into Equation (1):

Substitute these values into equation (1): $-2K + 2 - (2K + 2) - 2(4K + 2) = -4.$ Solving for $K$, we find: $K = \frac{1}{6}.$

Step 7: Calculating $\alpha + \beta + 6\gamma$:

Substitute $K = \frac{1}{6}$ into the expressions for $\alpha$, $\beta$, and $\gamma$: $\alpha = -2 \times \frac{1}{6} + 2 = \frac{10}{6} = \frac{5}{3},$ $\beta = 2 \times \frac{1}{6} + 2 = \frac{14}{6} = \frac{7}{3},$ $\gamma = 4 \times \frac{1}{6} + 2 = \frac{16}{6} = \frac{8}{3}.$ Therefore, $\alpha + \beta + 6\gamma = \frac{5}{3} + \frac{7}{3} + 6 \times \frac{8}{3} = \frac{24}{3} = 6.$

Answer: 6