2x^2+6x+5y+10=0 need vertex, directrex and focus,

Question

i only need to find the vertex, directrix and focus.. i can't quite understand this..

Answer 1

Given equation 2x ² + 6x + 5y +10 = 0

We first put the equation in to the form for a translated parabola (x - h )² = 4p (y - k )

To do this we complete the squre on the x  terms and move the other terms to right.

2x ² + 6x = -5y - 10

Here x² coefficient is 2, for perfect square make x ² coefficient 1 by divide each side by 2.

x ² + 3x = -5/2y - 5

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

x coefficient is 3,(half the x coefficient)² = 9/4

x ² + 3x + 9/4 = -5/2y - 5 + 9/4

(x + 3/2)² = - 5/2y - 11/4

In the next step we factored -5/2 from the right hand side to make the coefficient of our y  "+1", as this is the standard form.

(x + 3/2)² = -5/2(y + 11/10)

(x -( -3/2))² = 4(-5/8)(y - (-11/10))

Compare it to  parabola equation is (x - h )² = 4p (y - k ), where (h, k ) = vertex and p = directed distance from vertex to focus.

p = -5/8

Vertex of parabola  = (-3/2, -11/10)

Focus = (h , k + p ) = [-3/2, ((-11/10)-(5/8))] = (-3/2, -69/40)

Equation of directrix is y  = k - p

y  = (-11/10) - (-5/8)

y  = (-11/10) + (5/8)

y  = (-44+25)/40

y  = -19/40

Vertex (-3/2, -11/10), focus(-3/2, -69/40) and directrix  y  = -19/40.