7.2 Parabolas

Find the parabolas given the following conditions:

(a) the focus is [math](3,1)[/math] and the directrix is [math]y=3[/math].

(b) the focus is [math](-4,3)[/math] and the vertex is [math](5,3)[/math].

(c) the vertex is [math](2,2)[/math] and the directrix is [math]x=1[/math].

Find the foci/vertices and directrices of the following parabolas:

(a) [math]y^2=8x-4[/math]

(b) [math]2-x^2=4y[/math]

(c) [math]y=x^2-2x+3[/math]

(d) [math]4x=-y^2-6y+7[/math]

(Calculator) Light running parallel to the axis of symmetry of a parabola converges at its focus. Jordan has a parabolic solar cooker with a diameter of 20 inches and 25 inches deep at home. Today he wants to roast a goose using the solar cooker. Find the distance, in inches, between the bottom of the solar cooker and the goose such that the goose can be roasted by the sunlight.

Consider a parabola [math]\Gamma[/math]. Suppose that [math](3,2)[/math] and [math](3,-1)[/math] are the endpoints of its latus rectum. Find the equation of [math]\Gamma[/math].

Consider an “open left” parabola [math](y-b)^2=k(x-a)[/math] where [math]a, b[/math], and [math]k[/math] are constants. Which of the following statement(s) is/are true?

(Calculator) Consider a parabola [math]y^2=4x[/math] with vertex [math]O[/math] and focus [math]F[/math] as shown below.

[math]A, B, C[/math], and [math]D[/math] are on the parabola such that [math]\overline{AB}[/math] and [math]\overline{CD}[/math] are parallel to the [math]x[/math]-axis. [math]F[/math] is on [math]\overline{AB}[/math]. Let [math]a[/math] and [math]b[/math] be the areas of triangle [math]OAB[/math] and trapezoid [math]ABDC[/math] respectively. Given that [math]9a=b[/math], find the [math]x[/math]-coordinate of [math]C[/math].

Let [math]\Gamma[/math] be a parabola with focus [math](2,2)[/math] and directrix [math]y=-x[/math]. If the length of its latus rectum is [math]4\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?