Question

If two vertices of a triangle are $(5,-1)$ and $(-2,3)$ and if its orthocentre lies at the origin, then the coordinates of the third vertex are

• A. $(4,7)$
• B. $(-4,-7)$
• C. $(2,-3)$
• D. $(5,-1)$

Answer 1

The correct option is B $(-4,-7)$

$\because$ Orthocentre is intersection point of perpendicular drawn from the vertices of triangle.
Let $A\equiv (-2,3),B\equiv (5,-1)$
$\because$ Coordinates of orthocentre is $(0,0)$
Equation of line which passes through origin and perpendicular to $AB$ is
$y=-\frac{1}{\left(\frac{3+1}{-2-5}\right)}x$
$\Rightarrow 4y=7x...\left(1\right)$

Equation of the line which passes through $B$ and perpendicular to the line joining $A$ to the origin.i.e, BC side equation
$y+1=-\frac{1}{\left(\frac{3}{-2}\right)}(x-5)$
$\Rightarrow 3y+13=2x...\left(2\right)$

So, the coordinates of third vertex C is intersection point of $\left(1\right)$ and $\left(2\right)$.
i.e., $(-4,-7)$