Question

Through a point P(f, g, h) a plane is drawn at right angles to OP, to meet the axes in A, B, C. If OP = r, the
centroid of the triangle ABC is

• A. $(\frac{f}{3r},\frac{g}{3r},\frac{h}{3r})$
• B. $(\frac{r^2}{3f^2},\frac{r^2}{3g^2},\frac{r^2}{3h^2})$
• C. $(\frac{r^2}{3f},\frac{r^2}{3g},\frac{r^2}{3h})$
• D. $(\frac{r^2}{f},\frac{r^2}{g},\frac{r^2}{h})$

Answer 1

The correct option is C $(\frac{r^2}{3f},\frac{r^2}{3g},\frac{r^2}{3h})$

Equation of plane in normal form is
lx + my + nz = r . . . .(i)
Also, DR’s of OP are f, g, h
$\therefore$ DC’s of OP are $\frac{f}{r},\frac{g}{r},\frac{h}{r}FromEq.\left(i\right),\frac{f}{r}x+\frac{g}{r}y+\frac{h}{r}z=r\therefore \frac{x}{\left(\frac{r^2}{f}\right)}+\frac{y}{\left(\frac{r^2}{g}\right)}+\frac{z}{\left(\frac{r^2}{h}\right)}=1\Rightarrow \therefore A\equiv (\frac{r^2}{f},0,0);A\equiv (0,\frac{r^2}{g}0);C\equiv (0,0,\frac{r^2}{h})\therefore Centroidof\Delta ABCis(\frac{r^2}{3f},\frac{r^2}{3g},\frac{r^2}{3h})$