Question

A line segment joining a point A on x-axis to a point B on y-axis is such that AB = 15. If P is a point on AB such that \(\frac{AP}{PB} = \frac{2}{3}\), then the locus of P is

• A. \(x = 9\cos\theta, y = 6\sin\theta\)
• B. \(x = 6\cos\theta, y = 9\sin\theta\)
• C. \(x = 6\cos\theta, y = 6\sin\theta\)
• D. \(x = 9\cos\theta, y = 9\sin\theta\)

Hint:

For a rod of length \(L\) sliding on coordinate axes, the locus of a point dividing it in ratio \(m:n\) (from x-axis) is an ellipse \(\frac{x^2}{(nL/(m+n))^2} + \frac{y^2}{(mL/(m+n))^2} = 1\). Remember: the x-coordinate comes from the 'n' segment (adjacent to x-axis in cross multiplication logic).

Answer 1

The Correct Option is A

Step 1: Coordinate Setup:
Let point \(A\) be on the x-axis, \(A = (a, 0)\). Let point \(B\) be on the y-axis, \(B = (0, b)\). Given the length \(AB = 15\), we have the condition: \[ a^2 + b^2 = 15^2 = 225 \]
Step 2: Section Formula:
Let \(P(x, y)\) be the point dividing \(AB\) in the ratio \(AP:PB = 2:3\). Using the section formula for internal division: \[ P(x, y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \] Here \(m=2, n=3\), \(A(x_1, y_1) = (a, 0)\), and \(B(x_2, y_2) = (0, b)\). \[ x = \frac{2(0) + 3(a)}{2+3} = \frac{3a}{5} \implies a = \frac{5x}{3} \] \[ y = \frac{2(b) + 3(0)}{2+3} = \frac{2b}{5} \implies b = \frac{5y}{2} \]
Step 3: Finding the Locus:
Substitute \(a\) and \(b\) into the distance condition \(a^2 + b^2 = 225\): \[ \left(\frac{5x}{3}\right)^2 + \left(\frac{5y}{2}\right)^2 = 225 \] \[ \frac{25x^2}{9} + \frac{25y^2}{4} = 225 \] Divide the entire equation by 25: \[ \frac{x^2}{9} + \frac{y^2}{4} = 9 \] Divide by 9 to get the standard form of an ellipse: \[ \frac{x^2}{81} + \frac{y^2}{36} = 1 \] This describes an ellipse with semi-major axis \(A_{ellipse} = \sqrt{81} = 9\) and semi-minor axis \(B_{ellipse} = \sqrt{36} = 6\).
Step 4: Parametric Form:
The parametric coordinates for \(\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1\) are \(x = A\cos\theta, y = B\sin\theta\). Hence, \(x = 9\cos\theta, y = 6\sin\theta\).