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Q.

Triangle ABC has AB =20, BC =34 and CA =42. Point P lies on the circle with diameter AB. The greatest possible area of triangle APC is R, then which of the following options is (are) CORRECT?

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a

Number of positive integral divisors of R are 16.

b

Number of positive integral prime factors of R are 3. 

c

R is a three digit number.

d

Number of ways in which R can be written as product of perfect squares is 4

answer is A, B, C.

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Detailed Solution

Question Image

Area of triangle APC is greatest
 Height of the triangle is maximum
Maximum height of triangle APC = PQ = MP + MQ
M is mid point of AB  Centre of circle
MP = Radius of circle =  12(20)=10
MQ is largest  MQAM=BIABMQ=12hB
BT = Altitude of triangle through vertex B to side AC
hB=2Δb=2ΔAC 
Δ=(48)(28)(14)(6)=336 
 hB=2(336)42=16
 MQ=12hB=8
PQ = MP + MQ = 10 + 8 = 18
Maximum area of  Δle  APC=12×42×18=(21)(18)=378

R=378=2133.71
Number of positive integral divisors = (1 + 1) (3 + 1) (1 + 1)  = (2) (4) (2) = 16

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