Answer:
x^2=20/3(y)
Step-by-step explanation:
Given:
focus of parabola=(0,-5/3)
directrix y=5/3
Standard form of parabola :
(x - h)^2 = 4p (y - k),
where focus is:
(h, k + p)
directrix is :
y = k - p
Now equating the values we get
(0,-5/3)= (h,k+p)
h=0
k+p=-5/3
k=-5/3-p
Also
y=5/3 and y=k-p
i.e. k-p=5/3
Substituting k=-5/3-p in above we get:
-5/3-p-p=5/3
-2p=10/3
p=-5/3
Putting p=-5/3 in k-p=5/3 we get:
k-(-5/3)=5/3
k=5/3-5/3
k=0
Putting all the values in standard formula for parabola we get:
(x - (0))^2 = 4(-5/3) (y -(0))
x^2=-20/3(y) !
Answer: The required equation of the parabola is [tex] x^2=-\dfrac{20}{3}y.[/tex]
Step-by-step explanation: We are given to write the equation for a parabola with focus [tex]\left(0,-\dfrac{5}{3}\right)[/tex] and directrix [tex]y=\dfrac{5}{3}.[/tex]
Since the focus of the parabola lies on the y-axis, so the equation of the parabola is of the following form :
[tex](x-h)^2=4p(y-k)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
And, the directrix is [tex]y=k-p[/tex] and the focus is (h, k+p).
According to the given information, we have
[tex]k-p=\dfrac{5}{3}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)\\\\h=0,\\\\k+p=-\dfrac{5}{3}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)[/tex]
Adding equations (ii) and (iii), we get
[tex]2k=0\\\\\Rightarrow k=0[/tex]
and
[tex](ii)\Rightarrow 0-p=\dfrac{5}{3}\\\\\\\Rightarrow p=-\dfrac{5}{3}.[/tex]
Substituting the values of h, k and p in equation (i), we get
[tex](x-0)^2=4\times\left(-\dfrac{5}{3}\right)(y-0)\\\\\\\Rightarrow x^2=-\dfrac{20}{3}y.[/tex]
Thus, the required equation of the parabola is [tex] x^2=-\dfrac{20}{3}y.[/tex]
