JEE Advanced 2026 · Paper 2 · Mathematics Q7

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MathematicsMultiple correct+4 / -1Question 7 of 54

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 ?

The orthocentre of the triangle

PRT

\left(\dfrac{23}5,-4,\dfrac{31}5\right)

The orthocentre of the triangle

PRT

(4,3,5)

The area of the triangle

PRT

6\sqrt5

The area of the triangle

PRT

18\sqrt5

Answer and worked solution

Official answer: A, D

Foot $S$ . $L$ : $P+t(1,1,2)$ . With $\vec{PR}=(3,-3,6)$ , $t=\frac{3-3+12}{6}=2$ , so $S=(3,4,3)$ and $|PS|=2\sqrt6$ .

Point $T$ . $|ST|=2|PS|$ with $S$ between $P$ and $T$ , so $t=6$ : $T=(7,8,11)$ , $|PT|=6\sqrt6$ .

Area. $RS$ is the altitude from $R$ : $\vec{SR}=(1,-5,2)$ , $|SR|=\sqrt{30}$ .
$\text{Area}=\tfrac12\cdot6\sqrt6\cdot\sqrt{30}=18\sqrt5.$
(D) true, (C) false.

Orthocentre. It lies on the altitude $RS$ : $H=S+\lambda(1,-5,2)$ . It must also satisfy $\vec{PH}\cdot\vec{TR}=0$ with $\vec{TR}=(-3,-9,-6)$ :
$(2,2,4)\cdot(-3,-9,-6)+\lambda(1,-5,2)\cdot(-3,-9,-6)=-48+30\lambda=0\implies\lambda=\tfrac85.$
$H=\left(\frac{23}5,-4,\frac{31}5\right)$ . (A) true, (B) false.

Topics:Equation of a line in space Angles and distances Dot product

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JEE Advanced 2026 · Paper 2, Mathematics question 7