The locus of a point P which divides the line joining (1, 0) and `(2 cos theta, 2 sin theta)` internally in the ratio `2: 3` for all `theta`, is a

Options

A. straight line
B. circle
C. pair of straight line
D. parabola

Answer

To find the locus of the point \( P \) that divides the line segment joining the points \( (1, 0) \) and \( (2 \cos \theta, 2 \sin \theta) \) in the ratio \( 2:3 \), we can use the section formula.

### Step 1: Use the Section Formula
The coordinates of point \( P \) that divides the line segment joining points \( A(1, 0) \) and \( B(2 \cos \theta, 2 \sin \theta) \) in the ratio \( m:n = 2:3 \) can be calculated using the section formula:

\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

Substituting the values:
- \( m = 2 \)
- \( n = 3 \)
- \( A(1, 0) \) where \( x_1 = 1, y_1 = 0 \)
- \( B(2 \cos \theta, 2 \sin \theta) \) where \( x_2 = 2 \cos \theta, y_2 = 2 \sin \theta \)

### Step 2: Calculate the Coordinates of Point \( P \)
Using the section formula, we can find the coordinates of point \( P(h, k) \):

\[ h = \frac{2(2 \cos \theta) + 3(1)}{2 + 3} = \frac{4 \cos \theta + 3}{5} \]

\[ k = \frac{2(2 \sin \theta) + 3(0)}{2 + 3} = \frac{4 \sin \theta}{5} \]

### Step 3: Express \( \cos \theta \) and \( \sin \theta \) in terms of \( h \) and \( k \)
From the equations for \( h \) and \( k \):

1. For \( h \):
\[ 5h - 3 = 4 \cos \theta \quad \Rightarrow \quad \cos \theta = \frac{5h - 3}{4} \]

2. For \( k \):
\[ 5k = 4 \sin \theta \quad \Rightarrow \quad \sin \theta = \frac{5k}{4} \]

### Step 4: Use the Pythagorean Identity
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ \left(\frac{5h - 3}{4}\right)^2 + \left(\frac{5k}{4}\right)^2 = 1 \]

### Step 5: Simplify the Equation
Expanding the equation:

\[ \frac{(5h - 3)^2}{16} + \frac{(5k)^2}{16} = 1 \]

Multiplying through by 16:

\[ (5h - 3)^2 + (5k)^2 = 16 \]

### Step 6: Rearranging to Standard Form
This can be rearranged to:

\[ (5h - 3)^2 + (5k)^2 = 4^2 \]

### Step 7: Identify the Locus
This equation represents a circle with:
- Center at \( \left(\frac{3}{5}, 0\right) \)
- Radius \( \frac{4}{5} \)

### Conclusion
The locus of the point \( P \) is a circle centered at \( \left(\frac{3}{5}, 0\right) \) with a radius of \( \frac{4}{5} \).

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