Question

The line segment joining \( (5,0) \) and \( (10\cos\theta, 10\sin\theta) \) is divided internally in the ratio \(2:3\) at \(P\). If \(\theta\) varies, then the locus of \(P\) is

• A. a pair of straight lines
• B. a circle
• C. a straight line
• D. a parabola
• E. an ellipse

Hint:

Whenever trigonometric parameters appear as \(a\cos\theta, a\sin\theta\), expect a circle after elimination.

Answer 1

The Correct Option is B

Concept: If a point divides a line joining two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) internally in ratio \(m:n\), then its coordinates are: \[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

Step 1: Given points
\[ A = (5,0), \quad B = (10\cos\theta, 10\sin\theta) \] Ratio = \(2:3\)

Step 2: Coordinates of \(P\)
\[ x = \frac{2(10\cos\theta) + 3(5)}{5} = \frac{20\cos\theta + 15}{5} = 4\cos\theta + 3 \] \[ y = \frac{2(10\sin\theta) + 3(0)}{5} = \frac{20\sin\theta}{5} = 4\sin\theta \] Thus: \[ P = (4\cos\theta + 3,\; 4\sin\theta) \]

Step 3: Eliminate parameter
\[ x - 3 = 4\cos\theta,\quad y = 4\sin\theta \]

Step 4: Square and add
\[ (x-3)^2 + y^2 = (4\cos\theta)^2 + (4\sin\theta)^2 \] \[ = 16(\cos^2\theta + \sin^2\theta) = 16 \]

Step 5: Final equation
\[ (x-3)^2 + y^2 = 16 \]

Step 6: Interpretation
This is equation of a circle with: \[ \text{Centre} = (3,0), \quad \text{Radius} = 4 \]

Step 7: Final Answer
\[ \boxed{\text{a circle}} \]