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 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

$y = 2 a x$
C

$x + y = 0$
D

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

Solution

We know from geometry that the circumcentre, centroid and orthocentre of a triangle lie on a line. So the orthocentre of the triangle lies on the line joining the circumcentre $(0, 0)$ and the centroid $(\frac{(a + 1)^{2}}{2}, \frac{(a - 1)^{2}}{2})$ i.e. $\frac{(a + 1)^{2}}{2} y = \frac{(a - 1)^{2}}{2} x$
or $(a - 1)^{2} x - (a + 1)^{2} y = 0$ .

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