If $\mathrm{O}(0,0), \mathrm{A}(1,2)$ and $\mathrm{B}(3,4)$ are the vertices of triangle OAB , then the…

1. $3 x^2-x y-2 y^2=0$
2. $3 x^2+x y+2 y^2=0$
3. $3 x^2-x y+2 y^2=0$
4. $3 x^2+x y-2 y^2=0$

Solution

OD is the median $\begin{aligned} \therefore \quad & D \equiv\left(\frac{1+3}{2}, \frac{2+4}{2}\right) \\ & \Rightarrow D \equiv(2,3) \end{aligned}$
Equation of OD is $y=m x$ $\begin{aligned} & \Rightarrow y=\frac{3}{2} x \\ & \Rightarrow 3 x-2 y=0 \end{aligned}$
Slope of line $A B=\frac{2}{2}=1$ Given, $\mathrm{OE} \perp \mathrm{AB}$ $\therefore \quad$ Slope of $\mathrm{OE}=-1$ Equation of OE is $y=m x$ $\begin{aligned} & \Rightarrow y=-x \\ & \Rightarrow x+y=0 \end{aligned}$ $\therefore \quad$ Joint equation of median and altitude is $\begin{aligned} & (3 x-2 y)(x+y)=0 \\ & \Rightarrow 3 x^2+x y-2 y^2=0 \end{aligned}$

Asked in: MHT CET 2024 (04 May Shift 2)

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