JEE MAIN - Mathematics (2025 - 7th April Morning Shift - No. 5)
Let ABC be the triangle such that the equations of lines AB and AC 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 ABC , then the area of the triangle PBC is equal to
8
4
10
6
Explanation
Equation of line $A B$ is $3 y-x=2$
And $A C$ is $x+y=2$
In line $A B$,
When $y=0, x=-2$
$$\therefore \quad B(-2,0)$$
In line $A C$,
When $y=0, x=2$
$$\therefore \quad C(2 x, 0)$$
Equation of altitude of $B C$,
$$Y=x+2$$
Similarly, equation of altitude of $A B$,
$$y=-3 x+6$$
$\therefore$ On solving, orthocentre $P(1,3)$
$$\therefore \quad \operatorname{ar}(\triangle P B C)=6$$
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