find the vertex, focus, directrix, and the length of the focal chord for

Question

y^2+6y-2x+13=0

Answer 1

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

The parabola equation is y² + 6y - 2x + 13 = 0.

Separate y - variables and x - variables aside by adding 2x to each side.

y² + 6y + 13 = 2x.

To change the expression [y² + 6y ] into a perfect square trinomial add and subtract (half the y coefficient)²

 Here y coefficient = 6. so, (half the y coefficient)² = (6/2)²= 9.

Add 9 to each side.

y² + 6y + 9 + 13 = 2x + 9

(y + 3)² + 13 = 2x + 9

(y + 3)² = 2x + 9 - 13

(y + 3)² = 2x - 4

(y + 3)² = 2(x - 2)

(y - (- 3))² = 4 * (1/2)(x - 2).

Compare the above equation with standard form of the parabola : (y - k)² = 4p (x - h).

Vertex (h, k) = (2, - 3)

p = directed distance from vertex to focus = 1/2.

Focus is p distance to the left of vertex on the axis of symmetry.

Focus = (h+p, k) = (2 + 1/2, - 3) = (5/2, - 3).

Directrix = x = h - p = 2 - 1/2 = (4 - 1)/2 = 3/2.

Focal chord  : The chord passes through the focus.

The length of the focal chord is also equal to the length of the latus rectu.

Length of Latus Rectum = | 4p | units  = |4 * 1/2| = 2 units.

So, the length of the focal chord is 2 units.

Therefore vertex, focus and directrix and the length of the focal chord are (2, - 3) , (5/2, - 3), x = 3/2 and 2 units respectively.