Question

The coordinates of the orthocentre of the triangle formed by the lines $2x^2-3xy+y^2=0$ and $x+y=1$ is

• A. $(1,1)$
• B. $(\frac{1}{2},\frac{1}{2})$
• C. $(\frac{1}{3},\frac{1}{3})$
• D. $(\frac{1}{4},\frac{1}{4})$

Answer 1

The correct option is B $(\frac{1}{2},\frac{1}{2})$

$2x^2-3xy+y^2=0$

$t=\frac{y}{x}$

$\Rightarrow t^2-3t+2=0$

$\Rightarrow t=\frac{3\pm \sqrt{9-8}}{2}$

$\Rightarrow t=\frac{3\pm 1}{2}$

$\Rightarrow t=2,1$

And $x+y=1$ is third side.

Two sides of triangle
i.e. $y=x$ and $x+y=1$ are perpendicular to each other.

So, orthocenter of triangle is intersection of

$y=x$ & $x+y=1$ line

$\therefore y=\frac{1}{2}=x$

$(\frac{1}{2},\frac{1}{2})$ is orthocentre.