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 .

Explanation

To solve the problem of finding how far from Earth a body must be along a line towards the Sun so that the Sun's gravitational pull on it balances that of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Distance between the Earth and the Sun, \( r = 1.5 \times 10^{10} \) km.
- Mass of the Sun, \( M_s = 3.24 \times 10^5 \times M_e \) (where \( M_e \) is the mass of the Earth).

2. **Set Up the Problem:**
- Let the distance from the Earth to the body be \( x \).
- Therefore, the distance from the Sun to the body will be \( r - x \).

3. **Write the Gravitational Force Equations:**
- The gravitational force exerted by the Earth on the body is given by:
\[ F_e = \frac{G M_e m}{x^2} \]
- The gravitational force exerted by the Sun on the body is given by:
\[ F_s = \frac{G M_s m}{(r - x)^2} \]

4. **Set the Forces Equal for Balance:**
- For the gravitational pull to balance, we set \( F_e = F_s \):
\[ \frac{G M_e m}{x^2} = \frac{G M_s m}{(r - x)^2} \]
- The \( G \) and \( m \) cancel out:
\[ \frac{M_e}{x^2} = \frac{M_s}{(r - x)^2} \]

5. **Substitute the Mass of the Sun:**
- Substitute \( M_s = 3.24 \times 10^5 M_e \):
\[ \frac{M_e}{x^2} = \frac{3.24 \times 10^5 M_e}{(r - x)^2} \]
- Cancel \( M_e \):
\[ \frac{1}{x^2} = \frac{3.24 \times 10^5}{(r - x)^2} \]

6. **Cross-Multiply:**
- Cross-multiplying gives:
\[ (r - x)^2 = 3.24 \times 10^5 x^2 \]

7. **Expand and Rearrange:**
- Expanding the left side:
\[ r^2 - 2rx + x^2 = 3.24 \times 10^5 x^2 \]
- Rearranging gives:
\[ r^2 - 2rx + x^2 - 3.24 \times 10^5 x^2 = 0 \]
- This simplifies to:
\[ r^2 - 2rx - (3.24 \times 10^5 - 1)x^2 = 0 \]

8. **Use the Quadratic Formula:**
- This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where:
- \( a = -(3.24 \times 10^5 - 1) \)
- \( b = -2r \)
- \( c = r^2 \)
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

9. **Calculate the Values:**
- Substitute \( r = 1.5 \times 10^{10} \) km into the formula and solve for \( x \).

10. **Final Calculation:**
- After calculating, we find:
\[ x \approx 2.63 \times 10^7 \text{ km} \]
- To find the distance from the Sun:
\[ r - x \approx 1.5 \times 10^{10} - 2.63 \times 10^7 \approx 1.497 \times 10^{10} \text{ km} \]

### Final Answer:
The body must be approximately \( 2.63 \times 10^7 \) km from the Earth along the line towards the Sun.