In this exercise we have the general form of the equation of a circle, which is given by:
[tex]x^2+y^2+18x+4y+49=0[/tex]
The ordinary equation of a circle is:
[tex](x-h)^2+(y-k)^2=r^2 \\ \\ Developing \ this \ equation: \\ \\ x^2+y^2-2hk-2ky+h^2+k^2-r^2=0[/tex]
But this can be written as:
[tex]x^2+y^2+Dx+Ey+F=0 \\ \\ D=-2h, \ E=-2k, \ and \ F=h^2+k^2-r^2[/tex]
From the original equation we know that:
[tex]D=18, \ E=4, \ and \ F=49 \\ \\ So: \\ \\ h=-\frac{18}{2}=9, \ k=-\frac{4}{2}=2, \ and \ r=\sqrt{9^2+2^2-49}=6[/tex]
Finally:
[tex]\boxed{CENTER: \ (h,k)=(9,2)} \\ \\ \boxed{RADIUS: \ 6}[/tex]
We need to write this equation in the form:
[tex]x^2+y^2+Dx+Ey+F=0[/tex]
Since:
[tex]D=-2h, \ E=-2k, \ and \ F=h^2+k^2-r^2[/tex]
Then we need to find the center and radius in order to get D, E and F then. From the graph, it is easy to know that the center is [tex](h,k)=(-3,4)[/tex] and the radius is 2. Therefore:
[tex]D=-2(-3)=6, \ E=-2(4)=-8, \ and \ F=(-3)^2+(4)^2-(2)^2=21[/tex]
Finally, the general form of the equation is:
[tex]\boxed{x^2+y^2+6x-8y+21=0}[/tex]
A parabola is the set of all points in a plane that are equidistant from a fixed line called the directrix and a fixed point called the focus that does not lie on the line. Since the directrix is [tex]x=2[/tex] then this is a horizontal axis. So the standard for of the equation of a parabola that matches this form is:
[tex](y-k)^2=4p(x-h) \ \ p \neq 0[/tex]
The vertex is [tex](h,k)=(-5,8)[/tex] so our goal is to find [tex]p[/tex]:
[tex](y-8)^2=4p(x+5)[/tex]
We can find the absolute value of [tex]p[/tex] as follows:
[tex]\left|p\right|=2-(-5)=7[/tex]
Since the directrix is to the left of the vertex, the parabola opens to the left and hence:
[tex]p<0 \\ \\ So: \\ \\ p=-7[/tex].
Finally, the equation is:
[tex](y-8)^2=4(-7)(x+5) \\ \\ \boxed{(y-8)^2=-28(x+5)}[/tex]
Answer: The answer is x^2 + y^2 + 6x - 8y + 21 = 0.
Step-by-step explanation:
First find the original standard form equation of the circle.
So the circle has a center of (-3, 4) and a radius of 2. Write this in standard form:
(x + 3)^2 + (y - 4)^2 = 2^2
Simplify:
(x + 3)^2 + (y - 4)^2 = 4
(x^2 + 6x + 9) + (y^2 - 8y + 16) = 4
Gather like terms:
x^2 + y^2 + 6x - 8y + 9 + 16 - 4 = 0
x^2 + y^2 + 6x - 8y + 21 = 0
There you go!
