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Definition of Circle

If the line ...

Question

If the line $l x + m y + n = 0$ touches the circle $x^{2} + y^{2} = a^{2}$ , then prove that $(l^{2} + m^{2}) a^{2} = n^{2}$ .

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Solution

If the line $l x + m y + n = 0$ touches the circle $x^{2} + y^{2} = a^{2}$ , then length of the perpendicular from its centre $O (0, 0)$ is equal to its radius a.

Therefore,
$| \frac{l \times 0 + m \times 0 + n}{\sqrt{l^{2} + m^{2}}} | = a$

$(l^{2} + m^{2}) a^{2} = n^{2}$ , which is the required condition.

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