First slide

Cartesian plane

Question

If the circumcentre of a triangle lies at the origin and the centroid is the middle point of the line joining the point is $(a^{2} + 1, a^{2} + 1)$ and $(2 a, - 2 a)$ then the orthocentre satisfies the equation

Moderate

A

$y = (a^{2} + 1) x$
B

$y = 2 a x$
C

$x = y = 0$
D

$(a - 1)^{2} x - (a + 1)^{2} y = 0$

Solution

The circumcentre is at the origin and the centroid
is at $(\frac{a^{2} + 1 + 2 a}{2}, \frac{a^{2} + 1 - 2 a}{2})$ i.e at $(\frac{(a + 1)^{2}}{2}, \frac{(a - 1)^{2}}{2})$
Let $(x, y)$ be the coordinates of the orthocentre. Since centroid divides the line segment joining circumcentre and orthocentre in the ratio 1 : 2.
$∴ \frac{(a + 1)^{2}}{2} = \frac{x}{3} and \frac{a (a - 1)^{2}}{2} = \frac{y}{3}$
$\Rightarrow \frac{(a + 1)^{2}}{(a - 1)^{2}} = \frac{x}{y} \Rightarrow (a - 1)^{2} x = (a + 1)^{2} y$

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