The line joining ` (5, 0)` to `(10 cos theta, 10 sin theta ) ` is divided internally in the ratio ` 2 : 3 ` at P. the locus of P is

Answer

To find the locus of point \( P \) that divides the line segment joining the points \( (5, 0) \) and \( (10 \cos \theta, 10 \sin \theta) \) in the ratio \( 2:3 \), we can follow these steps:

### Step 1: Use the Section Formula
The coordinates of point \( P \) that divides the line segment joining points \( A(5, 0) \) and \( B(10 \cos \theta, 10 \sin \theta) \) in the ratio \( m:n = 2:3 \) can be found using the section formula:
\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
where \( (x_1, y_1) = (5, 0) \) and \( (x_2, y_2) = (10 \cos \theta, 10 \sin \theta) \).

### Step 2: Substitute the Values
Substituting the values into the section formula:
\[ P_x = \frac{2(10 \cos \theta) + 3(5)}{2 + 3} = \frac{20 \cos \theta + 15}{5} = 4 \cos \theta + 3 \]
\[ P_y = \frac{2(10 \sin \theta) + 3(0)}{2 + 3} = \frac{20 \sin \theta}{5} = 4 \sin \theta \]
Thus, the coordinates of point \( P \) are:
\[ P\left( 4 \cos \theta + 3, 4 \sin \theta \right) \]

### Step 3: Set Up the Equations
Let \( x = 4 \cos \theta + 3 \) and \( y = 4 \sin \theta \).

### Step 4: Express \( \cos \theta \) and \( \sin \theta \)
From the equation for \( x \):
\[ 4 \cos \theta = x - 3 \implies \cos \theta = \frac{x - 3}{4} \]
From the equation for \( y \):
\[ 4 \sin \theta = y \implies \sin \theta = \frac{y}{4} \]

### Step 5: Use the Pythagorean Identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \left( \frac{y}{4} \right)^2 + \left( \frac{x - 3}{4} \right)^2 = 1 \]
This simplifies to:
\[ \frac{y^2}{16} + \frac{(x - 3)^2}{16} = 1 \]

### Step 6: Multiply Through by 16
Multiplying through by 16 gives:
\[ y^2 + (x - 3)^2 = 16 \]

### Step 7: Rearranging the Equation
This represents a circle centered at \( (3, 0) \) with a radius of \( 4 \).

### Final Locus Equation
Thus, the locus of point \( P \) is given by:
\[ (x - 3)^2 + y^2 = 16 \]