Find the equations of the lines joining the origin to the points of intersection of the lines $y = m x + c$ with the circle $x^{2} + y^{2} = a^{2}$ and find the condition for these lines to be perpendicular.

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Solution

From the equation $\frac{y - m x}{c} = 1$

$∴ x^{2} + y^{2} = a^{2} \cdot 1^{2}$

or $x^{2} + y^{2} = a^{2} \frac{(y - m x)^{2}}{c^{2}}$

Ans. $(c^{2} - a^{2} m^{2}) x^{2} + 2 a m^{2} x y + (c^{2} - a^{2}) y^{2} = 0$

If it represents two perpendicular straight lines, then
coeff. of $x^{2} +$ coeff. of $y^{2} = 0$

$c^{2} - a^{2} m^{2} + c^{2} - a^{2} = 0$

$2 c^{2} = a^{2} (1 + m^{2})$ is the required condition.

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y = m x + c

x^{2} + y^{2} = a^{2}

y = m x + c

x^{2} + y^{2} = 2 a x + 2 b y

y = m x + c

x^{2} + y^{2} = a^{2}

x^{2} + y^{2} = a^{2} .

2 c^{2} = a^{2} (1 + m^{2}) .

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