How far from earth must a body be along a line towards the sun so that the sun's gravitational pull on it balances that of the earth . Distance between sun and earth's centre is `1.5 xx 10^(10)` km . Mass of the sun is `3.24 xx 10^(5)` times mass of the earth .

Answer

To find the distance from the Earth where the gravitational pull of the Earth and the Sun on a body is balanced, we can follow these steps:

### Step 1: Define the variables
Let:
- \( M_e \) = mass of the Earth
- \( M_s \) = mass of the Sun
- \( d \) = distance from the Earth to the Sun = \( 1.5 \times 10^{10} \) km
- \( x \) = distance from the Earth to the point where the gravitational forces balance

### Step 2: Set up the gravitational force equations
The gravitational force exerted by the Earth on the body is given by:
\[ F_e = \frac{G M_e m}{x^2} \]
where \( m \) is the mass of the body and \( G \) is the gravitational constant.

The gravitational force exerted by the Sun on the body is given by:
\[ F_s = \frac{G M_s m}{(d - x)^2} \]

### Step 3: Set the forces equal to each other
For the forces to balance, we set \( F_e = F_s \):
\[ \frac{G M_e m}{x^2} = \frac{G M_s m}{(d - x)^2} \]

### Step 4: Simplify the equation
We can cancel \( G \) and \( m \) from both sides:
\[ \frac{M_e}{x^2} = \frac{M_s}{(d - x)^2} \]

### Step 5: Substitute the mass of the Sun
Given that \( M_s = 3.24 \times M_e \), we substitute this into the equation:
\[ \frac{M_e}{x^2} = \frac{3.24 M_e}{(d - x)^2} \]

### Step 6: Cancel \( M_e \)
We can cancel \( M_e \) from both sides:
\[ \frac{1}{x^2} = \frac{3.24}{(d - x)^2} \]

### Step 7: Cross-multiply
Cross-multiplying gives:
\[ (d - x)^2 = 3.24 x^2 \]

### Step 8: Expand and rearrange
Expanding the left side:
\[ d^2 - 2dx + x^2 = 3.24 x^2 \]
Rearranging gives:
\[ d^2 - 2dx + x^2 - 3.24 x^2 = 0 \]
\[ d^2 - 2dx - 2.24 x^2 = 0 \]

### Step 9: Solve the quadratic equation
This is a quadratic equation in \( x \):
\[ -2.24 x^2 - 2dx + d^2 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = -2.24 \), \( b = -2d \), and \( c = d^2 \).

### Step 10: Calculate the values
Substituting \( d = 1.5 \times 10^{10} \) km into the formula will yield the value of \( x \).