Question

An iron rod of length $3$ units is such that its end points are always in contact with co-ordinate axes. Then locus of point of trisection of this rod contains the point

• A. $(1,0)$
• B. $(2,0)$
• C. $(0,1)$
• D. $(0,2)$

Answer 1

Answer: (A), (B), (C), (D)

$(0,2)$


Point of trisection divides a given line in the ratio of $2:1$ or $1:2$,

Case $1:$ Let $P$ divides $AB$ in the ratio $1:2.$
$\therefore (x_1,y_1)=(\frac{2h}{3},\frac{k}{3})$
$\Rightarrow h=\frac{3x_1}{2},k=3y_1$
$\overline{AB}=3\Rightarrow \frac{9x^2}{4}+9y^2=9$
$\Rightarrow$ Locus is $\frac{x^2}{4}+y^2=1$

Case $2:$ Let $P$ divides $AB$ in the ratio $2:1.$
$\therefore (x_1,y_1)=(\frac{h}{3},\frac{2k}{3})$
$\Rightarrow h=3x_1,k=\frac{3y_1}{2}$
$\overline{AB}=3\Rightarrow 9x^2+\frac{9y^2}{4}=9$
$\Rightarrow$ Locus is $x^2+\frac{y^2}{4}=1$

$\frac{x^2}{4}+y^2=1$ passess through $(2,0)$ and $(0,1).$

$x^2+\frac{y^2}{4}=1$ passess through $(1,0)$ and $(0,2).$