Respuesta :

Be breve bebe be hehe

Answer:

The coordinates of the focus are [tex](-3,0)[/tex]

Step-by-step explanation:

  • The general equation for any conic section is [tex]Ax^2+Bxy+Cy^2+Dx+Ey+F=0[/tex] where [tex]A[/tex], [tex]B[/tex], [tex]C[/tex], [tex]D[/tex], [tex]E[/tex], and [tex]F[/tex] are constants
  • If [tex]B^2-4AC<0[/tex], the conic section is either a circle or ellipse
  • If [tex]B^2-4AC=0[/tex], the conic section is a parabola
  • If [tex]B^2-4AC>0[/tex], the conic section is a hyperbola

Since [tex]B^2-4AC=0^2-4(1)(0)=0[/tex], the conic section is a parabola.

  • The standard form equation for a parabola is [tex](x-h)^2=4p(y-k)[/tex]
  • Vertex is [tex](h,k)[/tex]
  • Focus is [tex](h,k+p)[/tex]
  • Directrix is [tex]y=k-p[/tex]
  • Vertical axis is at the line [tex]x=h[/tex]
  • [tex]p\neq 0[/tex]

Convert general form into standard form by completing the square:

[tex]x^2+6x-4y+5=0[/tex]

[tex]x^2+6x+5=4y[/tex]

[tex]x^2+6x+9=4y+4[/tex]

[tex](x+3)^2=4(y+1)[/tex]

Now that the equation is in the form of [tex](x-h)^2=4p(y-k)[/tex], we can see that [tex]h=-3[/tex] and [tex]k=-1[/tex] which tells us that the vertex is at [tex](-3,-1)[/tex]. To determine the coordinates of the focus, we need to solve the equation [tex]4p=4[/tex] and plug the value of [tex]p=1[/tex] into [tex](h,k+p)[/tex] to get [tex](-3,-1+1)[/tex] which is [tex](-3,0)[/tex].

In conclusion, the coordinates of the focus for the conic section are [tex](-3,0)[/tex].

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