If the focus of a parabola divided a focal chord of the parabola in segment of length $3$ and $2$ the length of the latus rectum of the parabola is ?

Let $s(a,o)$ be the focus of the given parabola

Let the end points of the focal chord be $P\left( a{ t }^{ 2 },2at \right) \quad and\quad Q\left( \dfrac { a }{ { t }^{ 2 } } ,\dfrac { -2a }{ t } \right)$

$SP$ and $SQ$ are segments of the focal chord with length $b$ and $c$ respectively

therefore, $b=3$ and $c=2$

also, $SP=\sqrt { { \left( a-at \right) }^{ 2 }+4{ a }^{ 2 }{ t }^{ 2 } } =\quad a\left( 1+{ t }^{ 2 } \right) \\ SQ=\sqrt { { \left( a-\dfrac { a }{ { t }^{ 2 } } \right) }^{ 2 }+\dfrac { 4{ a }^{ 2 } }{ { t }^{ 2 } } } =\quad a\left( \dfrac { { t }^{ 2 }+1 }{ { t }^{ 2 } } \right)$

Now we have, $\dfrac { 1 }{ SP } +\dfrac { 1 }{ SQ } =\dfrac { 1 }{ a\left( 1+{ t }^{ 2 } \right) } +\dfrac { { t }^{ 2 } }{ a\left( 1+{ t }^{ 2 } \right) } =\dfrac { 1 }{ a } \\ \Rightarrow \dfrac { 1 }{ b } +\dfrac { 1 }{ c } =\dfrac { 1 }{ a } \\ \Rightarrow \dfrac { 1 }{ 3 } +\dfrac { 1 }{ 2 } =\dfrac { 1 }{ a } \\ \Rightarrow a=\dfrac { 6 }{ 5 }$