First slide

Definition of a circle

Question

The coordinates of the middle point of the chord intercepted on line $I x + m y + n = 0$ by the circle $x^{2} + y^{2} = a^{2}$ are

Moderate

A

$(\frac{- l}{l^{2} + m^{2}}, \frac{- m}{l^{2} + m^{2}})$
B

$(\frac{- \ln}{1^{2} + m^{2}}, \frac{- m n}{l^{2} + m^{2}})$
C

$(\frac{- l}{n (l^{2} + m^{2})}, \frac{- m}{n (l^{2} + m^{2})})$
D

none of these

Solution

Let $(x_{1}, y_{1})$ be the mid-point of the chord intercepted by the circle $x^{2} + y^{2} = a^{2}$ on the line $l x + m y + n = 0$ .
Then, the equation of the chord of the circle $x^{2} + y^{2} = a^{2}$
whose middle point is $(x_{1}, y_{1})$ is
$\begin{array}{l} x x_{1} + y y_{1} - a^{2} = {x_{1}}^{2} + {y_{1}}^{2} a^{2} \\ \Rightarrow x x_{1} + y y_{1} = {x_{1}}^{2} + {y_{1}}^{2} \end{array}$
Clearly, $l x + m y + n = 0$ and (i) represents the same line.
$∴ \frac{x_{1}}{l} = \frac{y_{1}}{m} = \frac{- (x_{1}^{2} + y_{1}^{2})}{n} = λ, say$
$\Rightarrow x_{1} = l λ, y_{1} = m λ$ and $x_{1}^{2} + y_{1}^{2} = - n λ$
$\begin{array}{l} \Rightarrow (l^{2} + m^{2}) λ^{2} = - n λ \\ \Rightarrow λ = - \frac{n}{l^{2} + m^{2}} \end{array}$
$∴ x_{1} = - \frac{\ln}{l^{2} + m^{2}}, y_{1} = \frac{m n}{l^{2} + m^{2}}$
Hence, the required points is $(\frac{- m}{l^{2} + m^{2}}, \frac{- m n}{l^{2} + m^{2}})$

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