Find the equation to the pair of straight lines joining the origin to the intersections of the straight line `y=mx + c` and the curve `x^2 + y^2=a^2` . Prove that they are at right angles if `2c^2=a^2(1+m^2)`.

Answer

To find the equation of the pair of straight lines joining the origin to the intersections of the straight line \( y = mx + c \) and the curve \( x^2 + y^2 = a^2 \), we can follow these steps:

### Step 1: Substitute the equation of the line into the equation of the circle
We start by substituting \( y = mx + c \) into the circle's equation \( x^2 + y^2 = a^2 \).

\[ x^2 + (mx + c)^2 = a^2 \]

### Step 2: Expand the equation
Expanding the left-hand side gives:

\[ x^2 + (m^2x^2 + 2mcx + c^2) = a^2 \]

Combining like terms results in:

\[ (1 + m^2)x^2 + 2mcx + (c^2 - a^2) = 0 \]

### Step 3: Identify the coefficients
This is a quadratic equation in \( x \), where:
- \( A = 1 + m^2 \)
- \( B = 2mc \)
- \( C = c^2 - a^2 \)

### Step 4: Use the quadratic formula
The roots of this equation (the \( x \)-coordinates of the intersection points) can be found using the quadratic formula:

\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Substituting our coefficients:

\[ x = \frac{-2mc \pm \sqrt{(2mc)^2 - 4(1 + m^2)(c^2 - a^2)}}{2(1 + m^2)} \]

### Step 5: Find the corresponding \( y \) values
For each \( x \), we can find the corresponding \( y \) using \( y = mx + c \).

### Step 6: Form the equation of the pair of lines
The pair of lines through the origin and the points of intersection can be expressed in the form:

\[ y = m_1x \quad \text{and} \quad y = m_2x \]

where \( m_1 \) and \( m_2 \) are the slopes corresponding to the roots of the quadratic equation.

### Step 7: Prove the lines are perpendicular
The lines are perpendicular if the product of their slopes \( m_1 \) and \( m_2 \) equals -1. This occurs when:

\[ m_1 m_2 = \frac{C}{A} = \frac{c^2 - a^2}{1 + m^2} \]

For the lines to be perpendicular, we need:

\[ m_1 m_2 = -1 \]

This leads to the condition:

\[ 2c^2 = a^2(1 + m^2) \]

### Conclusion
Thus, the equation of the pair of straight lines joining the origin to the intersections of the straight line and the curve is derived, and we have shown that they are at right angles under the given condition.

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