Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y 2 = 4 x with respect to the line x + y + 4 = 0 . If d denotes the distance between A and B ,

Question

Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y 2 = 4 x with respect to the line x + y + 4 = 0 . If d denotes the distance between A and B , and a denotes the area of △ S A B , where S is the focus of the parabola y 2 = 4 x , then the value of ( a + d ) is -

TestHub · Accepted Answer

To find image of P ( t 2 , 2 t ) ​ 1 x − t 2 ​ = 1 y − 2 t ​ = 1 2 + 1 2 − 2 ( t 2 + 2 t + 4 ) ​ = − ( t + 1 ) 2 − 3 x = t 2 − ( t + 1 ) 2 − 3 = − 2 t − 4 y = 2 t − ( t + 1 ) 2 − 3 = − t 2 − 4 t = 2 − x − 4 ​ ⇒ y + 4 = − ( 2 − x − 4 ​ ) 2 ⇒ ( y + 4 ) = − 4 ( x + 4 ) 2 ​ ⇒ x 2 = − 4 y ⇒ Focus ( − 4 , − 5 ) ​ Also, y = − 5 intersect ​ ∴ ( − 4 ) ( − 1 ) = ( x + 4 ) 2 4 = ( x + 4 ) 2 x + 4 = ± 2 x = − 2 , − 6 ⇒ d = 4 a = 2 1 ​ ​ 1 − 2 − 6 ​ 0 − 5 − 5 ​ 1 1 1 ​ ​ = 2 1 ​ ∣ [ 1 ( − 5 + 5 ) + 1 ( 10 − 30 )] ∣ = 2 1 ​ ( 20 ) a = 10 ∴ a + d = 14 ​

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