Find the equation to the pair of straight lines joining the origin to the intersections oi 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)`.

Explanation

To solve the problem, we need to find the equation of the pair of straight lines joining the origin to the intersections of the line \( y = mx + c \) and the circle \( x^2 + y^2 = a^2 \). We will also prove that these lines are at right angles under the condition \( 2c^2 = a^2(1 + m^2) \).

### Step 1: Substitute the line equation into the circle equation

We start by substituting \( y = mx + c \) into the circle equation \( x^2 + y^2 = a^2 \).

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