Post History
#3: Post edited
- 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
- 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
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.