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 rod, which is 3 cm from the end in contact with the x-axis.

Asked by Pragya Singh | 11 months ago |  113

1 Answer

Solution :-

Let AB be the rod making an angle Ɵ with OX and P(x,y) be the point on it such that

AP = 3cm.

Diagrammatic representation is as follows:

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 - Sarthaks eConnect  | Largest Online Education Community

Then, PB = AB – AP = (12 – 3) cm = 9cm [AB = 12cm]

From P, draw PQ ⊥ OY and PR ⊥ OX.

In ΔPBQ, cos θ = \( \dfrac{PQ}{QB}\)\( \dfrac{x}{9}\)

Sin θ =\( \dfrac{PR}{PA}\)\( \dfrac{y}{3}\)

we know that, sin2 θ +cos2 θ = 1,

So,

\( ( \dfrac{y}{3})^2\) + \( ( \dfrac{x}{9})^2\) = 1 or

\( \dfrac{x^2}{81}+\dfrac{y^2}{9}\) = 1

Answered by Abhisek | 11 months ago

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