Post History
#4: Post edited
by
Udi Fogiel
·
2022-03-21T21:21:24Z (about 2 years ago)
#3: Post edited
by
Udi Fogiel
·
2022-03-21T21:19:21Z (about 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
Udi Fogiel
·
2022-03-21T21:17:34Z (about 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
Udi Fogiel
·
2022-03-21T21:17:16Z (about 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.