Find the locus of point of intersection of 2 perpendicular tangents one to each of the two given confocals. Share with your friends Share 0 Manbar Singh answered this Let the 2 confocals bex2a2 + λ1 + y2b2 + λ1 = 1 ....1x2a2 + λ2 + y2b2 + λ2 = 1 ....2Now, the perpendicular tangents to conics 1 and 2 are : x cos α + y sin α = p1 .....3x sin α + y cos α = p2 ......4Now, from the condition of tangency a2 + λ1 cos2α + b2 + λ1 sin2α = p12 and a2 + λ2 sin2α + b2 + λ2 cos2α = p22 ⇒ a2cos2α + b2 sin2α + λ1 = p12 .....5and a2sin2α + b2 cos2α + λ2 = p22 .....6adding 5 and 6 a2cos2α + sin2α + b2cos2α + sin2α + λ1 + λ2 = p12 + p22⇒a2 + b2 + λ1 + λ2 = p12 + p22 .....7Now, the locus of the point of intersection of the tangents is obtained by eliminating α , p1 and p2.Squaring 3 and 4 and then adding, we get x2 + y2 =p12 + p22 ⇒ x2 + y2 = a2 + b2 + λ1 + λ2 Using 7This is required locus that is a circle. 0 View Full Answer
