The not-so-random tangent line meeting.

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We are given the curve (parabola) $y=ax^{2}$ , where $a\neq 0$ . Let us take $2$ random tangent lines to the given parabola, $t_{1}$ and $t_{2}$ , that have their points of tangency at the points $A$ and $B$ . These two tangent lines intersect at a certain point that we will call $C$ . The vertical line that goes through $C$ cuts the line segment $AB$ in the point $C'$ . If the value of $\frac{\overline{AC'}}{\overline{AB}}$ is $\frac{a}{b}$ , where $a$ and $b$ are two co-prime positive integers, what is the value of $a+b$ ?

Details and assumptions:

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

The answer is 3.

1 solution

Michael Mendrin
May 13, 2014

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

The not-so-random tangent line meeting.

The answer is 3.

1 solution

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