Let the point P of the focal chord PQ of the parabola y^2=16x be (1,-4).

Question

Let the point $P$ of the focal chord $PQ$ of the parabola $y^2=16x$ be $(1,-4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m:n$, $\gcd(m,n)=1$, then $m^2+n^2$ is equal to:

Accepted Answer

Here $a=4$; $P=(4t^2,8t)=(1,-4)\Rightarrow t=-\dfrac12$. For the focal chord, $Q$ has parameter $-1/t=2$: $Q=(16,16)$.

Focal distances: $PS=x_P+a=5,\ QS=x_Q+a=20$.

Focus divides $PQ$ in ratio $PS:SQ=5:20=1:4$, so $m=1,n=4$, and $m^2+n^2=17$.

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