![]() ![]() c 2 x 2 − 2 a 2 c x + a 4 = a 2 ( x 2 − 2 c x + c 2 + y 2 ) Expand the squares. c x − a 2 = − a ( x − c ) 2 + y 2 Divide by 4. 4 c x − 4 a 2 = − 4 a ( x − c ) 2 + y 2 Isolate the radical. 2 c x = 4 a 2 − 4 a ( x − c ) 2 + y 2 − 2 c x Combine like terms. ![]() x 2 + 2 c x + c 2 + y 2 = 4 a 2 − 4 a ( x − c ) 2 + y 2 + x 2 − 2 c x + c 2 + y 2 Expand remaining squares. x 2 + 2 c x + c 2 + y 2 = 4 a 2 − 4 a ( x − c ) 2 + y 2 + ( x − c ) 2 + y 2 Expand the squares. ( x + c ) 2 + y 2 = 2 a − ( x − c ) 2 + y 2 Move radical to opposite side. The ellipse is the set of all points ( x, y ) ( x, y ) such that the sum of the distances from ( x, y ) ( x, y ) to the foci is constant, as shown in Figure 5.ĭ 1 + d 2 = ( x − ( − c ) ) 2 + ( y − 0 ) 2 + ( x − c ) 2 + ( y − 0 ) 2 = 2 a Distance formula ( x + c ) 2 + y 2 + ( x − c ) 2 + y 2 = 2 a Simplify expressions. To derive the equation of an ellipse centered at the origin, we begin with the foci ( − c, 0 ) ( − c, 0 ) and ( c, 0 ). Deriving the Equation of an Ellipse Centered at the Origin Later we will use what we learn to draw the graphs. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. Later in the chapter, we will see ellipses that are rotated in the coordinate plane. That is, the axes will either lie on or be parallel to the x- and y-axes. In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Place the thumbtacks in the cardboard to form the foci of the ellipse. We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Each fixed point is called a focus (plural: foci). An ellipse is the set of all points ( x, y ) ( x, y ) in a plane such that the sum of their distances from two fixed points is a constant. This section focuses on the four variations of the standard form of the equation for the ellipse. The signs of the equations and the coefficients of the variable terms determine the shape. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. Conic sections can also be described by a set of points in the coordinate plane. ![]()
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