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.
Solution and Explanation
Let AB be the rod making an angle \(θ\) with OX and P (x, y) be the point on it such that AP = 3 cm.
Then, \(PB = AB - AP = (12 - 3) cm = 9 cm [AB = 12 cm] \)
From P, draw \(PQ⊥OY\) and \(PR⊥OX.\)
In \( \triangle PBQ, cos θ = \frac{PQ}{PB} = \frac{x}{9}\)
In \(\triangle PRA, Sin θ =\frac{ PR}{PA} =\frac{ y}{3}\)
Since, \(sin^2 θ +cos^2 θ = 1,\)
\((\frac{y}{3})^2 + (\frac{x}{9})^2 = 1\)
or \(\frac{x^2}{81} + \frac{y^2}{9} = 1\)
Thus, the equation of the locus of point P on the rod is \(\frac{x^2}{81} + \frac{y^2}{9} = 1\)
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Concepts Used:
Conic Sections
When a plane intersects a cone in multiple sections, several types of curves are obtained. These curves can be a circle, an ellipse, a parabola, and a hyperbola. When a plane cuts the cone other than the vertex then the following situations may occur:
Let ‘β’ is the angle made by the plane with the vertical axis of the cone
- When β = 90°, we say the section is a circle
- When α < β < 90°, then the section is an ellipse
- When α = β; then the section is said to as a parabola
- When 0 ≤ β < α; then the section is said to as a hyperbola
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