We are given the curve (parabola) , where . Let us take random tangent lines to the given parabola, and , that have their points of tangency at the points and . These two tangent lines intersect at a certain point that we will call . The vertical line that goes through cuts the line segment in the point . If the value of is , where and are two co-prime positive integers, what is the value of ?
Details and assumptions:
A point of tangency is the point at which a particular tangent line intersects its curve.
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L e t ( a , a 2 ) a n d ( b , b 2 ) b e t h e 2 r a n d o m t a n g e n t p o i n t s o n t h e p a r a b o l a y = x 2 . T h e n t h e t a n g e n t l i n e s a r e 2 a x − a 2 a n d 2 b x − b 2 , a n d t h e y i n t e r s e c t a t ( 2 1 ( a + b ) , a b ) . H e n c e , t h e l i n e x = 2 1 ( a + b ) p a s s e s h a l f w a y b e t w e e n t h e t w o t a n g e n t p o i n t s , a n d s o t h e a n s w e r i s 1 + 2 = 3 .