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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.

6.  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.

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Let \theta be the angle that rod makes with the ground,

Now, at a point 3 cm from the end,

\cos\theta=\frac{x}{9}

At the point touching the ground

\sin\theta=\frac{y}{3}

Now, As we know the trigonometric identity,

\sin^2\theta+\cos^2\theta=1

\left (\frac{x}{9} \right )^2+\left ( \frac{y}{3} \right )^2=1

\frac{x^2}{81}+\frac{y^2}{9}=1

Hence the equation is,

\frac{x^2}{81}+\frac{y^2}{9}=1

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