First slide

Cartesian plane

Question

Two vertices of a triangle are (3, – 2) and (–2, 3) and its orthocenter is (–6, 1). Then the third vertex of this triangle can NOT lie on the line

Moderate

A

4x + y = 2
B

5x + y = 2
C

3x + y = 3
D

6x + y = 0

Solution

Let $m_{1} = s l o p e o f$
$B H = \frac{1 - (- 2)}{- 6 - 3} = \frac{- 1}{3}$
$\Rightarrow$ Slope of $A C = 3$
$∴$ Equation of AC is
$y - 3 = 3 (x + 2)$ or $3 x - y + 9 = 0$ (1)
Let $m_{2} =$ slope of $C H = \frac{1 - 3}{- 6 + 2} = \frac{1}{2}$
$\Rightarrow$ Slope of AB $= - 2$
$∴$ Equation of AB is $y + 2 = - 2 (x - 3)$
or $2 x + y - 4 = 0$ (2)
As A lies on (1) and (2), coordinates of A are $(- 1, 6)$ and it does not lie on $5 x + y = 2$

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