Consider the circles S 1 : x 2 + y 2 + 2 x + 8 y - 23 = 0 and S 2 : x 2 + y 2 - 4 x + 10 y + 19 = 0 If the polars of the centre of a circle with respect to the another circle are L 1 and L 2 , then L 1 , L 2 are

Question

Consider the circles S 1 : x 2 + y 2 + 2 x + 8 y - 23 = 0 and S 2 : x 2 + y 2 - 4 x + 10 y + 19 = 0 If the polars of the centre of a circle with respect to the another circle are L 1 and L 2 , then L 1 , L 2 are, parallel and separated by a distance of 4 10 units, perpendicular and intersect at 1 , 3, perpendicular and intersect at 1 , - 5, parallel and separated by a distance of 2 10 units

MARKS App · Accepted Answer

We have S 1 : x 2 + y 2 + 2 x + 8 y - 23 = 0 c 1 - 1 , - 4 and S 2 : x 2 + y 2 - 4 x + 10 y + 19 = 0 c 2 2 , - 5 Given that the polars of the centre of a circle with respect to another circle are L 1 and L 2 We know that equation of polar is given by T = 0 Now L 1 : - x - 4 y - 2 x - 1 + 5 y - 4 + 19 = 0 L 1 : - 3 x + y + 1 = 0 L 2 : 2 x - 5 y + x + 2 + 4 y - 5 - 23 = 0 L 2 : - 3 x + y + 41 = 0 Clearly L 1 ∥ L 2 Distance between them is = 41 - 1 9 + 1 = 40 10 = 4 10 units.

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