Conic Sections: Parabolas

Transcription

1 Conic Sections: Parabolas Why are the graphs of parabolas, ellipses, and hyperbolas called 'conic sections'? Because if you pass a plane through a double cone, the intersection of the plane and the cone will be one of the 'conic sections'. The conic determined will depend on how the plane passes through the double cone. These drawings should illustrate this for you. Here is a double cone. The parabola is in black, the plane is parallel to the edge of the cone.

2 The ellipse is in black. If the plane is perpendicular to the axis of the double cone, the ellipse is a circle. The hyperbola is in black. The plane is parallel to the axis of the double cone. Parabola - geometric definition the set of points in a plane that are equidistant from a fixed point F (the focus) and a fixed line D (the directrix). We will derive the equation of a parabola with vertex at the origin using the above definition in such a way as to find the focus and directrix also.

3 From the definition and from the picture above, the two lengths, one from the focus to the point on ther parabola (x,y), the other from the point on the parabola (x,y) to the closest point on the directrix, together with the distance formula give us this what fact? ( x ) + ( y p) = ( x x) + [ y ( p) ] x + ( y p) = ( y + p) x + ( y p) = ( y + p) x + y y p + p = y + y p + p y x = p y = p x Definition focal length - distance from the focus to the vertex in a parabola Notes ) p is the 'focal length' in the formula y = p x ) If p>, the parabola opens up and if p< the parabola opens down. We could do the same analysis if the focus was at (p,) and directrix was x = -p to find that x = p y with the following: Notes: ) p is the 'focal length' in the formula x = p y ) if p>, the parabola opens to the right and if p< the parabola opens to the left.

4 Ex Find the focus and directrix of y x =. Here p =, so p =. The focus is at the point, and the directrix is y =. Ex Find the focus and directrix of y = x. Here p =, so p =. The focus is at the point, and the directrix is y =. 5 x x x

5 Ex Find the focus and directrix of the parabola and sketch the graph. 8x = y then try to think of a way to use a calculator to graph this one. x = y 8 p = p = 8 focus is (-,) directix is x =. How to sketch the graph using a calculator? First solve for y: there are two solutions y = 8x and y = 8x simplifying to y = x and y = x 5 x x 5 5 x

6 Ex Find the focus, vertex and directrix of the equation y + y + x + 5 =. If we can complete the square (we learned this in Precalc I when we learned about circles) we can get the graph into one of the following forms: y k = p ( x h) x h = ( y k) Vertex is at (h,k) p y + y = x 5 y + y + 9 = x ( y + 7) = x + ( y + 7) = ( x ) x = ( y + 7) vertex is at (,-7) p = focus is where? (,-7) directrix is where? x = x7 x x 6

7 Ex 5 find the equation of the parabola shown in the figure: y = ( p x + ) since we know the vertex, how do we find p? 5 = ( p + ) 8p = 6 6 p = 8 9 = 7 y = 9 7 ( x + ) y 7 = ( x + ) 6 Ex 6 Find the equation of the parabola: Focus is (-,-) directrix y =. Where is the vertex and what is the focal length?

8 Vertex is at the point, and p = y + = ( x + ) y + = ( x + ) 6 Books might give answer of 6y + = x 6x 9 and to use a GC, must solve for y: y = ( x + ) 6 ( x+ ) x Ex 7 Find an equation for the set of points in the xy plane that are equidistant from the point P(5,-) and the line L: y =. Vertex is at the point (5,) and p = -. y = ( x 5) Alternately, we could use the distance formula: Let (x,y) be on the parabola. Then ( x 5) + ( y + ) = ( x x) + ( y ) ( x 5) + ( y + ) = ( y ) ( x 5) + ( y + ) = ( y ) ( x 5) + y + y + = y 8y + 6

9 ( x 5) + y + = 8y + 6 ( x 5) = y + ( x 5) = ( y ) ( x 5) = ( y ) Ex 8 Find an equation of the parabola with vertical axis and passing through P(,-), Q(,-7) and R(-,). Sketch on a calculator to verify. We learned that a parabola is a graph of a polynomial of degree previously. Hence the equation must be of form y = a x + b x + c. Since the given points are on the parabola, we know that they must each satisfy the equation y = a x + b x + c for some a, b, and c yet to be determined. Using the given data, we can determine a system of equations whose solution will give us the a, b, and c. P is on the graph, so = a + b + c Q is on the graph, so 7 = a + b + c R is on the graph, so = a ( ) + b ( ) + c 9a + b + c = a + b + c = 7 a b + c = R + R, R + R R + R R, R 6

10 7 R + R 7 + R + R, R + R 5 y = x 5x 6 x 5x

11 P(,-), Q(,-7) and R(-,) 8 The reflective property of parabolas x is quite useful in the real world. Satellite dish, microwave tower receivers and many other things use this idea. Also if a light source is placed at the focus, then the light reflected by a parabolic mirror reflects away in parallel beams. Ex 9 If a dish has a foot in radius and the receiver is / of a square inch, how much is the signal amplified by the dish if all of the signal hitting the dish gets reflected to the receiver? Area of the dish in square inches is π sq inches and the area of the receiver was given as / sq inch. The ratio is π = π = so the signal is over 8 times stronger than it would be without the reflector.

12 Ex A flashlight mirror has the shape of a paraboloid (cross sections through the center are parabolas) and a diameter of inches and depth of inch. Where should the bulb be placed so that the light rays coming from the light are parallel to the axis of the paraboloid? Essentially, find the focus of this parabola x = p y 7 = p 7 = p p = 7 Place the bulb at the point 7,, or 7 inch from the bottom of the mirror.