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Let the sum of the focal distances of the point P (4,3) on the hyperbola H : x ^2 a ^2 - y ^2 ~b ^2 =1 be 8 5 3 . If for H , the length of the latus rectum is l and the product of the focal distances of the point P is m , then 9 l^2+6 ~m is equal to :-

Options

  1. A184
  2. B186
  3. C185
  4. D187

Correct answer

C. 185

Step-by-step solution

aligned & e x+a+e x-a=8 5 3 & 2 e x=8 5 3 & 2 e 4=8 5 3 & e= 5 3 & b^2=a^2 ( ( 5 3 )^2-1 ) & b^2= 2 3 a^2 aligned 16 a ^2 - 9 ~b ^2 =1 and b^2= 2 3 a^2 a ^2= 5 2 ~b ^2= 5 3 Now, aligned & = 2 ~b ^2 a & ^2= 4 ~b ^4 a ^2 & 9 ^2=36 25 9 5 2 & 9 ^2=40 & ~m =( ex + a )( ex - a ) & m = e ^2 x ^2- a ^2 & = 5 3 16- 5 2 = 145 6 aligned aligned & =6 ~m =145 & 9 ^2+6 ~m & 40+145=185 aligned option (3)

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