A rod of length $1.2\,\text m$ moves with its ends always touching the coordinate axes. The locus of a point $P$ on the rod, which is $0.3\,\text m$ from the end in contact with $x$-axis is an ellipse. Find the eccentricity.

A rod of length 1.2text m moves with its ends always touching the coordinate axes. The locus of a point P on t

Exercise 5.5 · Q7

1.2\,\text m

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Parametrise the rod by the angle it makes with the $x$ -axis, express $P$ 's coordinates in terms of that angle, and eliminate the angle to get an ellipse whose semi-axes are $(L-p)$ and $p$ .

Step 1. Set up. Let the rod make angle $\theta$ with the $x$ -axis; end on the $x$ -axis at $A=(L\cos\theta,0)$ , end on the $y$ -axis at $B=(0,L\sin\theta)$ , $L=1.2$ . Point $P$ is at distance $p=0.3$ from $A$ (along the rod towards $B$ ).

Step 2. Find $P$ 's coordinates. $P$ divides $AB$ so that $AP=p$ ; working through the geometry, $x=(L-p)\cos\theta$ and $y=p\sin\theta$ .

Step 3. Eliminate $\theta$ . …

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Q.A rod of length $1.2\,\text m$ moves with its ends always touching the coordinate axes. The locus of a point $P$ on the rod, which is $0.3\,\text m$ from the end in contact with $x$ -axis is an ellipse. Find the eccentricity.

Concept understanding — Real-life Applications of Conics

Q.A rod of length 1.2 m1.2\,\text m1.2m moves with its ends always touching the coordinate axes. The locus of a point PPP on the rod, which is 0.3 m0.3\,\text m0.3m from the end in contact with xxx-axis is an ellipse. Find the eccentricity.

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Q.A rod of length $1.2\,\text m$ moves with its ends always touching the coordinate axes. The locus of a point $P$ on the rod, which is $0.3\,\text m$ from the end in contact with $x$ -axis is an ellipse. Find the eccentricity.

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