Consider the triangle $OAB$ where $O \equiv (0,0),B(3,4)$. If orthocenter of triangle is $H(1,4)$, t

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Consider the triangle $OAB$ where $O \equiv (0,0),B(3,4)$. If orthocenter of triangle is $H(1,4)$, then coordinates of ' $A$ is :

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Sol. [Key Concept: With $O$ the origin and $B$ given, $A$ lies on the $y$-axis and $AH \perp OB$ locates the orthocentre $H$.]$BH\bot OA \Rightarrow A$ lies on $y$-axis equation of $AH$ is :$$y - 4 = - \frac{3}{4}(x - 1)$$Put $x = 0$,$$\begin{matrix} \Rightarrow & y = 4 + \frac{3}{4} = \frac{19}{4} \ \Rightarrow & A \equiv \left( 0,\frac{19}{4} ight) \end{matrix}$$

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