Let ABC be the triangle such that the equations of lines AB and AC be 3 y-x=2 and x+y=2, r…

Question

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

Accepted Answer

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 $P(1,3)$

$$\therefore \quad \operatorname{ar}(\triangle P B C)=6$$

Solutions · 1