The not-so-random tangent line meeting.

Level pending

We are given the curve (parabola) y = a x 2 y=ax^{2} , where a 0 a\neq 0 . Let us take 2 2 random tangent lines to the given parabola, t 1 t_{1} and t 2 t_{2} , that have their points of tangency at the points A A and B B . These two tangent lines intersect at a certain point that we will call C C . The vertical line that goes through C C cuts the line segment A B AB in the point C C' . If the value of A C A B \frac{\overline{AC'}}{\overline{AB}} is a b \frac{a}{b} , where a a and b b are two co-prime positive integers, what is the value of a + b a+b ?

Details and assumptions:

A point of tangency is the point at which a particular tangent line intersects its curve.


The answer is 3.

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1 solution

Michael Mendrin
May 13, 2014

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 ( 1 2 ( a + b ) , a b ) . H e n c e , t h e l i n e x = 1 2 ( 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. Let\quad (a,{ a }^{ 2 })\quad and\quad (b,{ b }^{ 2 })\quad be\quad the\quad 2\quad random\quad tangent\quad points\quad on\quad the\quad \\ parabola\quad y={ x }^{ 2 }.\quad Then\quad the\quad tangent\quad lines\quad are\quad 2ax-{ a }^{ 2 }\quad and\quad 2bx-{ b }^{ 2 },\quad \\ and\quad they\quad intersect\quad at\quad (\frac { 1 }{ 2 } (a+b),\quad ab).\quad \quad Hence,\quad the\quad line\quad \\ x=\frac { 1 }{ 2 } (a+b)\quad passes\quad halfway\quad between\quad the\quad two\quad tangent\quad points,\quad \\ and\quad so\quad the\quad answer\quad is\quad 1+2=3.\quad \\

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