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#4: Post edited by user avatar Udi Fogiel‭ · 2022-03-21T21:21:24Z (over 2 years ago)
#3: Post edited by user avatar Udi Fogiel‭ · 2022-03-21T21:19:21Z (over 2 years ago)
  • A parabola could be defined in the following way:
  • >parabola is a set of points, such that for any point $P$ of the set the distance $|PF|$ to a fixed point $F$, the focus, is equal to the distance $P\ell$ to a fixed line $\ell$, the directrix
  • In part one of the question we were asked to find the distance between a point on the parabola $D$, and the directrix. Because you found that the coordinates of $D$ are $(\frac{32}{p},8)$, and we know that the directrix of a canonical parabola of this form is the line $x=-\frac{p}{2}$, we can conclude that the distance between $D$ and the directrix is $\frac{p}{2}+\frac{32}{p}$.
  • From part two, as toy mentioned, we can infer that the distance between $D$ and the focus $F$ is the sum of the radii of the circles, which is $p+4+\frac{p}{2}$. But from the definition of a parabola those two lengths are equal, so we can conclude that
  • $$
  • p+4+\frac{p}{2}=\frac{p}{2}+\frac{32}{p}\ \Rightarrow\ (p+8)(p-4)=p^2+4p-32=0
  • $$
  • It is given that $p>0$, so the only concise solution is $p=4$ as you got as well.
  • A parabola could be defined in the following way:
  • >parabola is a set of points, such that for any point $P$ of the set, the distance $|PF|$ to a fixed point $F$, the focus, is equal to the distance $P\ell$ to a fixed line $\ell$, the directrix
  • In part one of the question we were asked to find the distance between a point on the parabola $D$, and the directrix. Because you found that the coordinates of $D$ are $(\frac{32}{p},8)$, and we know that the directrix of a canonical parabola of this form is the line $x=-\frac{p}{2}$, we can conclude that the distance between $D$ and the directrix is $\frac{p}{2}+\frac{32}{p}$.
  • From part two, as you mentioned, we can infer that the distance between $D$ and the focus $F$ is the sum of the radii of the circles, which is $p+4+\frac{p}{2}$. But from the definition of a parabola those two lengths are equal, so we can conclude that
  • $$
  • p+4+\frac{p}{2}=\frac{p}{2}+\frac{32}{p}\ \Rightarrow\ (p+8)(p-4)=p^2+4p-32=0
  • $$
  • It is given that $p>0$, so the only concise solution is $p=4$ as you got as well.
#2: Post edited by user avatar Udi Fogiel‭ · 2022-03-21T21:17:34Z (over 2 years ago)
  • A parabola could be defined in the following way:
  • >parabola is a set of points, such that for any point $P$ of the set the distance $|PF|$ to a fixed point $F$, the focus, is equal to the distance $P\ell$ to a fixed line $\ell$, the directrix
  • In part one of the question we were asked to find the distance between a point on the parabola $D$, and the directrix. Because you found that the coordinates of $D$ are $(\frac{32}{p},8)$, and we know that the directrix of a canonical parabola of this form is the line $x=-\frac{p}{2}$, we can conclude that the distance between $D$ and the directrix is $\frac{p}{2}+\frac{32}{p}$.
  • From part two, as toy mentioned, we can infer that the distance between $D$ and the focus $F$ is the sum of the radii of the circles, which is $p+4+\frac{p}{2}$. But from the definition of a parabola those two lengths are equal, so we can conclude that
  • $$
  • p+4+\frac{p}{2}=\frac{p}{2}+\frac{32}{p}\ \Rightarrow\ (p+8)(p-4)=p^2+4p-32=0
  • $$
  • It is givven that $p>0$, so the only concise solution is $p=4$ as you got as well.
  • A parabola could be defined in the following way:
  • >parabola is a set of points, such that for any point $P$ of the set the distance $|PF|$ to a fixed point $F$, the focus, is equal to the distance $P\ell$ to a fixed line $\ell$, the directrix
  • In part one of the question we were asked to find the distance between a point on the parabola $D$, and the directrix. Because you found that the coordinates of $D$ are $(\frac{32}{p},8)$, and we know that the directrix of a canonical parabola of this form is the line $x=-\frac{p}{2}$, we can conclude that the distance between $D$ and the directrix is $\frac{p}{2}+\frac{32}{p}$.
  • From part two, as toy mentioned, we can infer that the distance between $D$ and the focus $F$ is the sum of the radii of the circles, which is $p+4+\frac{p}{2}$. But from the definition of a parabola those two lengths are equal, so we can conclude that
  • $$
  • p+4+\frac{p}{2}=\frac{p}{2}+\frac{32}{p}\ \Rightarrow\ (p+8)(p-4)=p^2+4p-32=0
  • $$
  • It is given that $p>0$, so the only concise solution is $p=4$ as you got as well.
#1: Initial revision by user avatar Udi Fogiel‭ · 2022-03-21T21:17:16Z (over 2 years ago)
A parabola could be defined in the following way:

>parabola is a set of points, such that for any point $P$  of the set the distance $|PF|$ to a fixed point $F$, the focus, is equal to the distance $P\ell$ to a fixed line $\ell$, the directrix

In part one of the question we were asked to find the distance between a point on the parabola $D$, and the directrix. Because you found that the coordinates of $D$ are $(\frac{32}{p},8)$, and we know that the directrix of a canonical parabola of this form is the line $x=-\frac{p}{2}$, we can conclude that the distance between $D$ and the directrix is $\frac{p}{2}+\frac{32}{p}$.

From part two, as toy mentioned, we can infer that the distance between $D$ and the focus $F$ is the sum of the radii of the circles, which is $p+4+\frac{p}{2}$. But from the definition of a parabola those two lengths are equal, so we can conclude that 
$$
p+4+\frac{p}{2}=\frac{p}{2}+\frac{32}{p}\ \Rightarrow\ (p+8)(p-4)=p^2+4p-32=0
$$      
It is givven that $p>0$, so the only concise solution is $p=4$ as you got as well.