If the lines joining the origin to the points of intersection of the lines `y=mx+c` will the circle `x^(2)+y^(2)=a^(2)` be perpendicular then the required condition is……………

Answer

To solve the problem, we need to find the condition under which the lines joining the origin to the points of intersection of the line \( y = mx + c \) with the circle \( x^2 + y^2 = a^2 \) are perpendicular.

### Step-by-Step Solution:

1. **Equation of the Circle**:
The given equation of the circle is:
\[ x^2 + y^2 = a^2 \]

2. **Substituting the Line Equation**:
Substitute the line equation \( y = mx + c \) into the circle equation. This gives:
\[ x^2 + (mx + c)^2 = a^2 \]
Expanding this, we get:
\[ x^2 + (m^2x^2 + 2mcx + c^2) = a^2 \]
Simplifying further:
\[ (1 + m^2)x^2 + 2mcx + (c^2 - a^2) = 0 \]

3. **Finding the Points of Intersection**:
The above equation is a quadratic in \( x \). The points of intersection can be found using the quadratic formula:
\[ x = \frac{-2mc \pm \sqrt{(2mc)^2 - 4(1 + m^2)(c^2 - a^2)}}{2(1 + m^2)} \]

4. **Condition for Perpendicularity**:
For the lines joining the origin to the points of intersection to be perpendicular, the condition is that the sum of the coefficients of \( x^2 \) and \( y^2 \) in the homogeneous form of the equation must equal zero.

The coefficients from the equation derived earlier are:
- Coefficient of \( x^2 \): \( 1 + m^2 \)
- Coefficient of \( y^2 \): \( 1 \)

Therefore, we need:
\[ (1 + m^2) + 1 = 0 \]
However, this is not possible since \( 1 + m^2 \) is always positive.

5. **Revising the Condition**:
Instead, we need to analyze the coefficients of the quadratic equation:
\[ (1 + m^2)x^2 + 2mcx + (c^2 - a^2) = 0 \]
The condition for the lines to be perpendicular is given by:
\[ (2mc)^2 - 4(1 + m^2)(c^2 - a^2) = 0 \]

6. **Solving the Condition**:
Expanding the above condition:
\[ 4m^2c^2 - 4(1 + m^2)(c^2 - a^2) = 0 \]
Simplifying gives:
\[ 4m^2c^2 - 4c^2 - 4m^2a^2 + 4a^2 = 0 \]
Rearranging leads to:
\[ 4(m^2 - 1)c^2 + 4a^2(1 - m^2) = 0 \]
Thus, we have:
\[ 2c^2 = a^2(1 + m^2) \]

### Final Condition:
The required condition is:
\[ 2c^2 = a^2(1 + m^2) \]