Let A(a, b), B(3,4) and C(-6,-8) respectively denote the centroid, circumcentre and orthoc…

Question

Let $A(a, b), B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2 a+3,7 b+5)$ from the line $2 x+3 y-4=0$ measured parallel to the line $x-2 y-1=0$ is

Accepted Answer

$$\mathrm{A}(\mathrm{a}, \mathrm{b}), \quad \mathrm{B}(3,4), \quad \mathrm{C}(-6,-8)$$

$$\Rightarrow \mathrm{a}=0, \mathrm{~b}=0 \quad \Rightarrow \mathrm{P}(3,5)$$

Distance from $$\mathrm{P}$$ measured along $$\mathrm{x}-2 \mathrm{y}-1=0$$

$$\Rightarrow x=3+r \cos \theta, \quad y=5+r \sin \theta$$

$$\begin{aligned}
& \text { Where } \tan \theta=\frac{1}{2} \
& \mathrm{r}(2 \cos \theta+3 \sin \theta)=-17 \
& \Rightarrow \mathrm{r}=\left|\frac{-17 \sqrt{5}}{7}\right|=\frac{17 \sqrt{5}}{7}
\end{aligned}$$

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