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  • The coordinates of the middle point of the chord intercepted on line <img alt="\mathrm{l x+m y+n=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bl%20x&plus;m%20y&plus;n

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

Option: 1

\mathrm{\left(\frac{-l}{l^2+m^2}, \frac{-m}{l^2+m^2}\right)}


Option: 2

\mathrm{\left(\frac{-l n}{l^2+m^2}, \frac{-m n}{l^2+m^2}\right)}


Option: 3

\mathrm{\left(\frac{-l}{n\left(l^2+m^2\right)}, \frac{-m}{n\left(l^2+m^2\right)}\right)}


Option: 4

\mathrm{\text { none of these }}


Answers (1)

best_answer

Let \mathrm{\left(x_1, y_1\right)} be the mid-point of the chord intercepted by the circle \mathrm{x^2+y^2=a^2} on the line \mathrm{l x+m y+n=0.}

Then, the equation of the chord of the circle \mathrm{x^2+y^2=a^2} whose middle point is \mathrm{\left(x_1, y_1\right)} is

\mathrm{ \begin{aligned} & 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{aligned} }                  ..(i)

Clearly, \mathrm{l x+m y+n=0} and (i) represents the same line.

\mathrm{ \begin{aligned} & \therefore \quad \frac{x_1}{l}=\frac{y_1}{m}=\frac{-\left(x_1^2+y_1^2\right)}{n}=\lambda \text { (say) } \\\\ & \Rightarrow x_1=l \lambda, y_1=m \lambda \text { and } x_1^2+y_1^2=-n \lambda \\\\ & \Rightarrow \quad\left(l^2+m^2\right) \lambda^2=-n \lambda \Rightarrow \lambda=-\frac{n}{l^2+m^2} \\\\ & \therefore \quad x_1=-\frac{l n}{l^2+m^2}, y_1=\frac{-m n}{l^2+m^2} \end{aligned} }

Hence, the required point is \mathrm{\left(\frac{-l n}{l^2+m^2}, \frac{-m n}{l^2+m^2}\right)}

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Deependra Verma

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