Question

A conic (E) has equation y²+2y+8x+9=0

a) State the nature of (E)

b) Determine its characteristic elements

c) Sketch (E) indicating all the elements.

d) Find the length of the lactus.

Answer 1

a)

$$E: y^2+2y+8x+9=0$$

$$y^2+2y+1+8x+8=0$$

A conic (E) is a parabola. Vertex form:

$$x=-\dfrac{1}{8}(y+1)^2-1$$

b)

$p=-2, k=-1, h=-1$

Vertex: $(h,k)=(−1,−1)$

Focus: $(h+p,k)=(-3,-1)$

Eccentricity: $1$

Directrix: $x=h-p, x=1$

Latus rectum: $x=−3$

The length of the latus rectum: $8$

Axis of symmetry: $y=−1$

Focal parameter: $1-(-3)=4$

x-intercept: $y=0, x=-\dfrac{1}{8}(0+1)^2-1=-\dfrac{9}{8}$

 $(−9/8,0)$

No y-intercepts.

(c)

d) Latus rectum: $x=−3$

Find ends of latus rectum

$$-3=-\dfrac{1}{8}(y+1)^2-1$$

$$y+1=\pm4$$

$$(-3, -5), (-3, 3)$$

The length of the latus rectum

$$3-(-5)=8$$