Let a conic C pass through the point ( 4 , − 2 ) and P ( x , y ) , x ≥ 3 , be any point on C . Let the slope of the line touching the conic C only at a single point P be half the slope of the line joi

Question

Let a conic C pass through the point ( 4 , − 2 ) and P ( x , y ) , x ≥ 3 , be any point on C . Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and ( 3 , − 5 ) . If the focal distance of the point ( 7 , 1 ) on C is d , then 12 d equals ______

TestHub · Accepted Answer

P ( x , y ) & x ≥ 3 Slope of line at P ( x , y ) will be d x d y ​ = 2 1 ​ ( x − 3 y + 5 ​ ) ​ ⇒ 2 ( y + 5 ) d y ​ = ( x − 3 ) 1 ​ d x ⇒ 2 ln ( y + 5 ) = ln ( x − 3 ) + C ​ Passes through ( 4 , − 2 ) ​ ⇒ 2 ln ( 3 ) = ln ( 1 ) + C ⇒ C = 2 ln ( 3 ) ⇒ 2 ln ( y + 5 ) = ln ( x − 3 ) + 2 ln ( 3 ) ⇒ 2 ( ln ( 3 y + 5 ​ ) ) = ln ( x − 3 ) ⇒ ( 3 y + 5 ​ ) 2 = ( x − 3 ) ​ ⇒ ( y + 5 ) 2 = 9 ( x − 3 ) Parabola 4 a = 9 a = 4 9 ​ ​ ​ d = ( 4 7 ​ ) 2 + 6 2 ​ d = 4 625 ​ ​ d = 4 25 ​ 12 d = 75 ​

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