First slide

Equation of line in Straight lines

Question

If a vertex of an equilateral triangle is the origin and the side opposite to it has the equation x + y = 1 then the orthocentre of the triangle is

Moderate

A

$(\frac{1}{3}, \frac{1}{3})$
B

$(\frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3})$
C

$(\frac{2}{3}, \frac{2}{3})$
D

None of these

Solution

$Clearly orthocentre ' H' lies on the line x - y = 0$
$Now distance of O (0, 0) from the line x + y - 1 = 0 is \frac{1}{\sqrt{2}} .$
$C e n t r o i d d i v i d e s m e d i a n i n t h e r a t i o 2 : 1 ∴ O H = \frac{2}{3} \cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{3} (since triangle is equilateral, centroid coincides with orthocentre)$
$∴ orthocentre \equiv (\frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3})$

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