Question

State the necessary and sufficient condition for lx+my+n=0 to be a normal to the circle
x²+ y²+ 2gx+2fy+C =0​

Answer 1

We have to state the necessary and sufficient condition for lx + my + n = 0 to be a normal to the circle x² + y² + 2gx + 2fy + c = 0.

solution : any line will be normal to the circle only if line passing through centre of the circle.

equation of circle is x² + y² + 2gx + 2fy + c = 0

so centre of circle = (-g, - f)

so equation of line must satisfy the centre of circle.

i.e., l(-g) + m(-f) + n = 0

⇒lg + mf = n

Therefore the required condition is lg + mf = n.

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