We will begin thistime with our "directrix" circle, our focus, and ourtangent line already constructed.Ģ. The only difference will be that we chooseour focus to be outside the circle.ġ. We will construct a hyperbolain Geometer's Sketchpad using the same methods we used for theellipse construction.
![gsp5 moveable points gsp5 moveable points](https://cdn.mysagestore.com/d8b7f87687a11c6d55e595bdbc0d1c3b/contents/9102410023/9102410023.jpg)
To investigate this constructionyourself, click here.Ī hyperbola can be definedas the set of all points the difference of whose distances fromtwo foci is a constant. It's because the focuswe chose is closer to the center of the circle! It looks the two fociwill always be the original focus and the center of the circle.Why does this ellipse appear to be more circular-shaped than ourprevious one? Is this true for any focus inside thecircle? Let's find out.ĥ. It is interestingto note that our two foci turn out to be the original focus andthe center of the circle. We can constructthe set of all tangent lines by moving the movable point aroundthe circle and tracing the tangent line.Ĥ. Once again, this perpendicularbisector is one of the tangent lines to the ellipse. Following the sameprocedure as the parabola construction, we will construct a movablepoint on the circle, a segment from the movable point to the focus,and the perpendicular bisector of that segment.ģ.
![gsp5 moveable points gsp5 moveable points](https://i.ytimg.com/vi/bNvRnQOu0R8/maxresdefault.jpg)
We will begin withour "directrix" circle and a focus.Ģ. Only this time,we will let a circle act as our "directrix." For thisconstruction, we will assume that the focus lies inside the circle.ġ. This construction willbe much like the construction for the parabola. We are going to constructan ellipse using Geometer's Sketchpad. With this one pointon the parabola, we can construct the set of all of points bysimply moving the movable point along the directrix and tracingour point on the parabola.Īnd, here is our parabola!To investigate this construction yourself, clickhere.Īn ellipse can be definedas the set of all points whose sum of its distance from two fociis a constant. So, if we draw a perpendicular through the movable pointto intersect our original tangent line, the intersection willgive us one point that lies on the parabola.ħ. The distance from a point to a lineis measured by drawing the perpendicular through the point tothe line. With this set of tangentlines, how can we find the points that actually lie on the parabola?įor that, we will haveto remember that every point on a parabola is equidistant fromthe directrix and the focus. With this one tangentline, we can construct the set of all of the tangent lines tothe parabola by simply moving the movable point along the directrixand tracing our tangent line.Ħ. This perpendicular bisector willrepresent one of the tangent lines to the graph of the parabola.ĥ. Using that midpoint,we will draw a perpendicular bisector through the segment fromthe movable point to the focus. In order to find apoint that is equidistant from the movable point and the focus,we will find the midpoint of the segment that connects them.Ĥ. Next, we will makea movable point on the directrix line to help us trace the pathof our parabola.ģ. We will begin withour directrix line and focus.Ģ. For our construction, wewill assume that the focus does not lie on the directrix line.ġ.
![gsp5 moveable points gsp5 moveable points](http://img.usdarts.com/images/stmp02dt.jpg)
We are going to constructa parabola using Geometer's Sketchpad.
![gsp5 moveable points gsp5 moveable points](https://i.ytimg.com/vi/1IUIRylW70w/maxresdefault.jpg)
In other words, for any pointon a parabola, the length of the green line is equal to the lengthof the blue line. Explorations of conic sections with sketchpad ConstructingConic Sections by:Lauren WrightĬonic sections are formedby the intersection of parallel planes and a double cone - formingellipses, parabolas, and hyperbolas, respectively.Ī parabola can be definedas the set of points equidistant from a line, called the directrix,and a fixed point, called the focus.