Locus of the image of the point ( 2 , 3 ) in the line 2 x - 3 y + 4 + k x - 2 y + 3 = 0 , k ∈ R , is a

Question

Locus of the image of the point ( 2 , 3 ) in the line 2 x - 3 y + 4 + k x - 2 y + 3 = 0 , k ∈ R , is a, Circle of radius 3, Straight line parallel to x-axis., Straight line parallel to y-axis., Circle of radius 2

MARKS App · Accepted Answer

Let image is ( α , β ) ∴ point ( 2 + α 2 , 3 + β 2 ) lies on the given variable line. ∴ 2 ( 2 + α 2 ) − 3 ( 3 + β 2 ) + 4 + k ( 2 + α 2 − 2 × 3 + β 2 + 3 ) = 0 ⇒ 2 α − 3 β + 3 + k ( α − 2 β + 2 ) = 0 . . . . . . . 1 again, product of two slopes = − 1 ⇒ 3 − β 2 − α × 2 + k 3 + 2 k = − 1 ⇒ 6 − 2 β + k ( 3 − β ) = − ( 6 − 3 α ) − k ( 4 − 2 α ) ⇒ k ( 7 − 2 α − β ) = 3 α + 2 β − 12 ⇒ k = 3 α + 2 β − 12 7 − 2 α − β Substituting value of k in , 1 we get 2 α − 3 β + 3 + 3 α + 2 β + − 12 7 − 2 α − β ( α − 2 β + 2 ) = 0 ⇒ 14 α − 4 α 2 − 2 α β − 21 β + 6 α β + 3 β 2 + 21 − 6 α − 3 β + 3 α 2 − 6 α β + 6 α + 2 α β − 4 β 2 + 4 β − 12 α + 24 β − 24 = 0 ⇒ α 2 + β 2 − 2 a − 4 β + 3 = 0 ∴ Locus is x 2 + y 2 − 2 x − 4 y + 3 = 0 Which is a circle with radius = 1 + 4 − 3 = 2

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