First slide

Cartesian plane

Question

The circumcentre of a triangle lies at the origin and the centroid is the midpoint of the line segment joining the points $(a^{2} + 1, a^{2} + 1)$ and $(2 a, - 2 a)$ , then the orthocentre lies on the line.

Moderate

A

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

$y = 2 ax$
C

$x + y = 0$
D

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

Solution

We know that, circumcentre, centroid and orthocentre of a triangle are collinear.
Hence, orthocentre lies on the line joining circumcentre (0, 0) and centroid $(\frac{(a + 1)^{2}}{2}, \frac{(a - 1)^{2}}{2})$
Equation of the line is
$\begin{array}{l} \frac{(a + 1)^{2}}{2} y = \frac{(a - 1)^{2}}{2} x \\ (a - 1)^{2} x - (a + 1)^{2} y = 0 \end{array}$
Therefore, the equation of the line where the orthocentre lies is $(a - 1)^{2} x - (a + 1)^{2} y = 0$

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