Question

If $\triangle ABC$ is right angled at A, where $A\equiv(4,2,x)$, $B\equiv(3,1,8)$ and $C\equiv(2,-1,2)$, then the value of $x$ is

• A. 4
• B. 2
• C. 3
• D. 1

Hint:

Geometry Tip: When dealing with angles in 3D coordinate geometry, converting the points to vectors and utilizing the dot product is almost always the fastest method.Two vectors are perpendicular if and only if their dot product is 0.

Answer 1

The Correct Option is C

Concept: 3D Geometry - Direction Ratios and Dot Product.

Step 1: Identify the geometric constraint. Since $\triangle ABC$ is right-angled at vertex A, the side vectors originating from A must be strictly perpendicular. Mathematically, this means the dot product of vector $\overline{AB}$ and vector $\overline{AC}$ must be exactly equal to zero ($\overline{AB}\cdot\overline{AC}=0$).

Step 2: Calculate vector $\overline{AB}$. The vector from point $A(4,2,x)$ to point $B(3,1,8)$ is found by subtracting the coordinates of A from B: $\overline{AB} = (3-4)\hat{i} + (1-2)\hat{j} + (8-x)\hat{k} = -\hat{i} - \hat{j} + (8-x)\hat{k}$.

Step 3: Calculate vector $\overline{AC}$. Similarly, the vector from point $A(4,2,x)$ to point $C(2,-1,2)$ is: $\overline{AC} = (2-4)\hat{i} + (-1-2)\hat{j} + (2-x)\hat{k} = -2\hat{i} - 3\hat{j} + (2-x)\hat{k}$.

Step 4: Set up the dot product equation. Apply the condition $\overline{AB}\cdot\overline{AC}=0$. Multiply the corresponding $\hat{i}$, $\hat{j}$, and $\hat{k}$ components and sum them: $(-1)(-2) + (-1)(-3) + (8-x)(2-x) = 0$. This simplifies to $2 + 3 + (16 - 10x + x^2) = 0$.

Step 5: Solve the resulting quadratic equation. Combine the constant terms: $x^2 - 10x + 21 = 0$. Factor the quadratic equation: $(x-3)(x-7) = 0$. This gives two possible values for $x$: $x=3$ or $x=7$. Looking at the given options, only $3$ is available. $$ \therefore \text{The correct value of } x \text{ is } 3. $$