Circles and Ellipses
Introduction to Circles
Derive Equation of a Circle from Pythagorean Theorem
Domain and Range of Circles
Circles at Origin
Circles with translations
Circles (mixed set)
Circles (Mixed and Semi-Circles)
Circles (manipulation of equation, graphing and applications)
Applications of Circles
Introduction to Ellipses
Identify Characteristics of Ellipses
Domain and Range of Ellipses
Finding the Equation of Ellipse
Graphing Ellipses
Manipulate Ellipse Equations
Applications of Ellipses
Typically we are given the equation of an ellipse in the standard form $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(x−h)2a2​+(y−k)2b2​=1 or $\frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$(x−h)2b2​+(y−k)2a2​=1 where $a>b$a>b. Ideally we would like to express any equation of an ellipse in such a way, since we can immediately identify its centre and the length of the semi-major and semi-minor axes:
$\left(h,k\right)$(h,k) is the centre of the ellipse.
$a$a is the length of semi-major axis.
$b$b is the length of semi-minor axis.
But generally speaking, the equation of an ellipse can be expressed in many ways. For instance, the following pair of equations both represent the same ellipse.
$9\left(x-2\right)^2+4\left(y-3\right)^2=36$9(x−2)2+4(y−3)2=36 or $9x^2-36x+4y^2-24x=-36$9x2−36x+4y2−24x=−36
In order to express the above equations in standard form, we can manipulate both sides by multiplying or by completing the square.
Consider the equation of the ellipse $x^2-12x+36y^2=0$x2−12x+36y2=0
Rewrite the equation in the standard form $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(x−h)2a2​+(y−k)2b2​=1 where $a>b$a>b.
Think: Completing the square allows us to rewrite the quadratic expressions in $x$x as a perfect square of the form $\left(x-h\right)^2$(x−h)2. Notice that the quadratic expression in $y$y is already a perfect square.
To complete the square in $x$x we look at the expression $x^2-12x$x2−12x, halve and square the coefficient of $x$x, and then add the result to both sides of the equation.
Do: Halving and squaring $-12$−12 gives a result of $36$36, so we add this result to both sides of the equation:
$x^2-12x+36+36y^2=36$x2−12x+36+36y2=36
So altogether we get the following steps:
$x^2-12x+36y^2$x2−12x+36y2 $=$= $0$0 (Writing down the equation)
$x^2-12x+36+36y^2$x2−12x+36+36y2 $=$= $36$36 (Completing the square for $x$x)
$\left(x-6\right)^2+36y^2$(x−6)2+36y2 $=$= $36$36 (Factorising the perfect square)
$\frac{\left(x-6\right)^2}{36}+y^2$(x−6)236​+y2 $=$= $1$1 (Dividing both sides by $36$36)
The equation of the ellipse is then $\frac{\left(x-6\right)^2}{36}+y^2=36$(x−6)236​+y2=36. Optionally, the standard form can be made more obvious by writing it as $\frac{\left(x-6\right)^2}{6^2}+\frac{y^2}{1^2}=36$(x−6)262​+y212​=36.
Reflect: The equation of the ellipse centred at $\left(6,0\right)$(6,0) with a vertical semi-major axis of length $6$6 units and a horizontal semi-minor axis of length $1$1 unit can be expressed by the equation $x^2-12x+36y^2=0$x2−12x+36y2=0 or the equation $\frac{\left(x-6\right)^2}{36}+y^2=1$(x−6)236​+y2=1.
Consider the equation of the ellipse $9x^2-36x+4y^2-24x=-36$9x2−36x+4y2−24x=−36.
Rewrite the equation in the standard form $\frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$(x−h)2b2​+(y−k)2a2​=1 where $a>b$a>b.
Think: Completing the square allows us to rewrite the terms containing $x$x and $y$y in the form $\left(x-h\right)^2$(x−h)2 and $\left(y-k\right)^2$(y−k)2. Before we do so, it will be more convenient if we factorise so that the coefficients of $x^2$x2 and $y^2$y2 are $1$1.
Do: In this case, the coefficient of $x^2$x2 is $9$9 the coefficient of $y^2$y2 is $4$4:
$9\left(x^2-4x\right)+4\left(y^2-6x\right)=-36$9(x2−4x)+4(y2−6x)=−36
Now we can complete the square in $x$x and $y$y:
$9\left(x^2-4x+4\right)+4\left(y^2-6x+9\right)=-36+9\times4+4\times9$9(x2−4x+4)+4(y2−6x+9)=−36+9×4+4×9
Putting this together, we get the following steps:
$9x^2-36x+4y^2-24x$9x2−36x+4y2−24x $=$= $-36$−36 (Writing down the equation)
$9\left(x^2-4x\right)+4\left(y^2-6x\right)$9(x2−4x)+4(y2−6x) $=$= $-36$−36 (Factorising the leading coefficients of each variable)
$9\left(x^2-4x+4\right)+4\left(y^2-6x+9\right)$9(x2−4x+4)+4(y2−6x+9) $=$= $-36+9\times4+4\times9$−36+9×4+4×9 (Completing the square for $x$x and $y$y)
$9\left(x^2-4x+4\right)+4\left(y^2-6x+9\right)$9(x2−4x+4)+4(y2−6x+9) $=$= $36$36 (Simplifying the constant terms)
$9\left(x-2\right)^2+4\left(y-3\right)^2$9(x−2)2+4(y−3)2 $=$= $36$36 (Rewriting the perfect squares in factorised form)
$\frac{\left(x-2\right)^2}{4}+\frac{\left(y-3\right)^2}{9}$(x−2)24​+(y−3)29​ $=$= $1$1 (Dividing both sides by $36$36)
So the equation of the ellipse can be expressed in the form $\frac{\left(x-2\right)^2}{4}+\frac{\left(y-3\right)^2}{9}=1$(x−2)24​+(y−3)29​=1.
Reflect: The centre of the ellipse is $\left(h,k\right)=\left(2,3\right)$(h,k)=(2,3), the length of the vertical semi-major axis is $3$3 units and the length of the horizontal semi-minor axis is $2$2 units. From this, we can draw the equation of the ellipse as shown below.
Graph of the ellipse $9x^2-36x+4y^2-24x=-36$9x2−36x+4y2−24x=−36.
Consider the ellipse with equation $\left(x+3\right)^2+4\left(y-1\right)^2=16$(x+3)2+4(y−1)2=16.
Write the equation in standard form $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(x−h)2a2​+(y−k)2b2​=1 where $a>b$a>b.
Draw the graph of the ellipse.
The graph of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$x29​+y24​=1 is given below.
Write $4x^2+9\left(y-2\right)^2=36$4x2+9(y−2)2=36 in the standard form $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(x−h)2a2​+(y−k)2b2​=1 where $a>b$a>b.
What transformation of the ellipse drawn gives the ellipse $4x^2+9\left(y-2\right)^2=36$4x2+9(y−2)2=36?
Translated to the right by $2$2 units
Translated to the left by $2$2 units
Translated up by $2$2 units
Translated down by $2$2 units
Consider the ellipse with equation $4x^2+24x+5y^2+20y+36=0$4x2+24x+5y2+20y+36=0.
What are the coordinates of the centre of the ellipse?
What are the coordinates of the vertices?
Apply the geometry of conic sections
Apply the geometry of conic sections in solving problems