Question

A rod of length l slides with ends on two perpendicular lines. Find the locus of its mid point

Answer 1

Answer:

A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the ...

Answer 2

AnswEr:

Let two perpendicular lines be the coordinate axes. Let AB be a rod of length l. Let the coordinates of A and B be (a,0) and (0,b) respectively. As the rod slides the values of a and b change. So a and b are two variables. Let P(h,k) be the mid-point of the rod AB in one of the infinite positions it attains. Then,

$\qquad \tt \: h = \frac{a + 0}{2} \: \: \: and \: \: \: k = \frac{0 + b}{2} \\ \\ \hookrightarrow \tt \: h = \frac{a}{2} \: \: \: and \: \: \: k = \frac{b}{2} \: - - - (i)$

From ∆OAB, we have

$\rm \qquad {ab}^{2 } = {ob}^{2} + {ob}^{2} \\ \\ \implies \tt \: {a}^{2} + {b}^{2} = {l}^{2} \\ \\ \implies \tt \: {(2h)}^{2} + {(2k)}^{2} = {l}^{2} \qquad \tt \: (using \: i) \\ \\ \implies \tt \: 4 {h}^{2} + 4 {k}^{2} = {l}^{2}$

Hence, the locus of (h,k) is 4x² + 4y² = l².