First slide

Cartesian plane

Question

If the circumcentre of the triangle formed by the points $(0, 0), (2, - 1)$ and $(- 1, 3)$ is at $(\frac{5}{2}, \frac{5}{2})$ then the coordinates of its orthcentre are

Moderate

A

$(- 4, - 3)$
B

$(4, 3)$
C

$(- 4, 3)$
D

none of these

Solution

The coordinates of the centroid are $G (\frac{0 + 2 - 1}{3}, \frac{0 - 1 + 3}{3})$ i.e. $(\frac{1}{3}, \frac{2}{3})$ . Let $(α, β)$ be the coordinates of orthocentre H. The coordinates of circumcentre are $O^{'} (\begin{array}{ll} 5 5 \\ 2 2 \end{array})$ .
Since G divides O' H the ratio 1 : 2.
$∴ \frac{α + 5}{3} = \frac{1}{3}$ and $\frac{β + 5}{3} = \frac{2}{3} \Rightarrow α = - 4, β = - 3$

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