Question

Let $L$ be the straight line joining the points $P(1, 2, -1)$ and $Q(2, 3, 1)$. Let $S$ be the foot of the perpendicular drawn from the point $R(4, -1, 5)$ to the line $L$. Another line passing through $R$ intersects $L$ at a point $T$ such that the point $S$ divides the line segment $PT$ internally in the ratio $|PS| : |ST| = 1 : 2$, where $|PS|$ and $|ST|$ are the lengths of the line segments $PS$ and $ST$, respectively. Then which of the following statements is (are) TRUE?

• A. The orthocentre of the triangle $PRT$ is $\left( \frac{23}{5}, -4, \frac{31}{5} \right)$
• B. The orthocentre of the triangle $PRT$ is $(4, 3, 5)$
• C. The area of the triangle $PRT$ is $6\sqrt{5}$
• D. The area of the triangle $PRT$ is $18\sqrt{5}$

Hint:

In a triangle where the foot of the altitude from one vertex is known on the opposite side, the orthocentre is easily found by parameterizing the altitude line and using the dot product with another side vector.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This 3D geometry problem involves finding coordinates of specific points relative to a line and then analyzing properties of a triangle formed by these points.

Step 2: Key Formula or Approach:

• Equation of line through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$.

• Foot of perpendicular using dot product of direction vectors.

• Section formula for internal division.

• Orthocentre is the intersection of altitudes.

Step 3: Detailed Explanation:

• Line $L$ passes through $P(1,2,-1)$ and $Q(2,3,1)$. Direction vector $\vec{d} = (1, 1, 2)$.

• Equation of $L: \vec{r} = (1,2,-1) + \lambda(1,1,2)$.

• Point $S$ on $L$: $S = (1+\lambda, 2+\lambda, -1+2\lambda)$. Vector $\vec{RS} = (\lambda-3, \lambda+3, 2\lambda-6)$.

• $\vec{RS} \cdot \vec{d} = 0 \implies (\lambda-3) + (\lambda+3) + 2(2\lambda-6) = 0 \implies 6\lambda - 12 = 0 \implies \lambda = 2$.

• So, $S = (3, 4, 3)$. Length $|PS| = \sqrt{(3-1)^2 + (4-2)^2 + (3-(-1))^2} = \sqrt{4+4+16} = \sqrt{24} = 2\sqrt{6}$.

• $S$ divides $PT$ in $1:2$ ratio: $\vec{S} = \frac{1\vec{T} + 2\vec{P}}{3} \implies \vec{T} = 3\vec{S} - 2\vec{P}$.
\[ T = 3(3,4,3) - 2(1,2,-1) = (9,12,9) - (2,4,-2) = (7, 8, 11). \]
• $|PT| = \sqrt{(7-1)^2 + (8-2)^2 + (11-(-1))^2} = \sqrt{36+36+144} = \sqrt{216} = 6\sqrt{6}$.

• Altitude $|RS| = \sqrt{(4-3)^2 + (-1-4)^2 + (5-3)^2} = \sqrt{1+25+4} = \sqrt{30}$.

• Area of $\triangle PRT$: Base is $PT$, height is $RS$ (since $RS \perp L$ and $P, T$ are on $L$). \[ \text{Area} = \frac{1}{2} \times 6\sqrt{6} \times \sqrt{30} = 3\sqrt{180} = 3 \times 6\sqrt{5} = 18\sqrt{5}. \] Statement (D) is TRUE, (C) is FALSE.

• Orthocentre $H$: Lies on altitude $RS$. $H = (3+k, 4-5k, 3+2k)$ (dir of $RS$ is $(1,-5,2)$).
Altitude from $P$ to $RT$ must pass through $H$. Vector $\vec{PH} = (2+k, 2-5k, 4+2k)$ is perp to $\vec{RT} = (3, 9, 6)$.
\[ 3(2+k) + 9(2-5k) + 6(4+2k) = 0 \implies 6+3k + 18-45k + 24+12k = 0 \implies -30k + 48 = 0 \] \[\implies k = 1.6 = \frac{8}{5}. \] \[ H = \left( 3 + \frac{8}{5}, 4 - 8, 3 + \frac{16}{5} \right) = \left( \frac{23}{5}, -4, \frac{31}{5} \right). \] Statement (A) is TRUE.

Step 4: Final Answer:
The true statements are (A) and (D).