Mathematics · Straight lines

JEE Main 2025 — 7 April, Morning Shift — Question 28

Let $A B C$ be the triangle such that the equations of lines $A B$ and $A C$ be $3 y-x=2$ and $x+y=2$ , respectively, and the points $B$ and $C$ lie on $x$ -axis. If $P$ is the orthocentre of the triangle $A B C$ , then the area of the triangle $P B C$ is equal to

Option A:
4
Option B:
6
Correct
Option C:
8
Option D:
10

Answer: B

Step-by-step solution

AB: 3y-x=2, \qquad AC: x+y=2

$A(1,1)$ by solving $3y-x=2$ and $x+y=2$

On $AB$ , $y=0 \Rightarrow x=-2 \Rightarrow B(-2,0)$

On $AC$ , $y=0 \Rightarrow x=2 \Rightarrow C(2,0)$

Slope of $BC=0 \Rightarrow$ altitude from $A:\ x=1$

Slope of $AC=-1 \Rightarrow$ altitude from $B:\ y=x+2$

Solving $x=1,\ y=x+2$ :

P(1,3)

\text{Area}(\triangle PBC) =\tfrac12 \times BC \times \text{height} =\tfrac12 \times 4 \times 3 =6

\boxed{\operatorname{ar}(\triangle PBC)=6}

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 7 April, Morning Shift
Subject: Mathematics
Chapter: Straight lines
Topic: Special Points in a Triangle