Class 11

Math

Co-ordinate Geometry

Conic Sections

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.

$AP=3$ cm and $PB=12−3=9$ cm

we can draw perpendiculars from $P$ to X-axis and Y-axis at point $R$ and $Q$ respectively.

Then, $PQ=x$ and $PR=y$

Please refer to video forthe diagram.

Now, in $ΔPBQ$,

$PBPQ =cosθ$

$⇒9x =cosθ→(1)$

Now, in $ΔAPR$,

$PAPR =sinθ$

$⇒3y =sinθ→(2)$

Now, squaring and adding (1) and (2),

$81x_{2} +9y_{2} =cos_{2}θ+sin_{2}θ$

$⇒81x_{2} +9y_{2} =1$

,which is the required equation.