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

##### 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:

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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