A = ( − 3 , 0 ) and B = ( 3 , 0 ) are the extremities of the base A B of triangle P A B . If the vertex P varies such that the internal bisector of angle A PB of the triangle divides the opposite side

Question

A = ( − 3 , 0 ) and B = ( 3 , 0 ) are the extremities of the base A B of triangle P A B . If the vertex P varies such that the internal bisector of angle A PB of the triangle divides the opposite side A B into two segments A D and B D such that A D : B D = 2 : 1 , then the maximum value of the length of the altitude of the triangle drawn through the vertex P is

TestHub · Accepted Answer

The point E dividing A B externally in the ratio 2 : 1 is ( 9 , 0 ) . Since P lies on the circle described on D E as diameter, coordinates of P are of the form ( 5 + 4 cos θ , 4 sin θ ) ∴ maximum length of the altitude drawn from P to the base A B = ∣4 sin θ ∣ m a x ​ = 4

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