Question 67 of 70intermediate🔧 ApplyLong Answer4 marks

A rod of length 1212 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 33 cm from the end in contact with the x-axis.

Correct Answer

The equation of the locus of point P is:

x281+y29=1\frac{x^2}{81} + \frac{y^2}{9} = 1

which is an ellipse.

Exercise: Miscellaneous Exercise on Chapter 10 | Q: 5 | (Page 29)
For More Understanding

Explanation

This problem is solved using the same method shown in Example 19 of the textbook context. The key insight is to express the coordinates of point P in terms of the angle θ\theta that the rod makes with the x-axis.

By using trigonometric ratios from the right triangles formed by dropping perpendiculars from P to both axes, and applying the fundamental identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, we can eliminate the parameter θ\theta to obtain the locus equation.

Solution Steps

  1. Step 1: Let AB be the rod of length 1212 cm with end A on the x-axis and end B on the y-axis. Let P(x,y)P(x, y) be a point on the rod such that AP=3AP = 3 cm (distance from end in contact with x-axis).

  2. Step 2: Calculate PB=AB−AP=12−3=9PB = AB - AP = 12 - 3 = 9 cm.

  3. Step 3: Let θ\theta be the angle the rod makes with the x-axis. Draw perpendiculars PQ and PR from P to the y-axis and x-axis respectively.

  4. Step 4: From triangle PBQ, cos⁡θ=x9\cos\theta = \frac{x}{9} (since PB=9PB = 9 cm).

  5. Step 5: From triangle PRA, sin⁡θ=y3\sin\theta = \frac{y}{3} (since AP=3AP = 3 cm).

  6. Step 6: Using the identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, substitute the values:

    (y3)2+(x9)2=1\left(\frac{y}{3}\right)^2 + \left(\frac{x}{9}\right)^2 = 1

  7. Step 7: Simplify to get:

    x281+y29=1\frac{x^2}{81} + \frac{y^2}{9} = 1

    which represents an ellipse.

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