In the equation of an ellipse, 4 x^2+9 y^2-16 x+18 y-11=0, determine the standard form of the eq

In the equation of an ellipse, 4 x^2+9 y^2-16 x+18 y-11=0,...

Key Concepts

Conic Sections

Conic sections are curves that are formed by the intersection of a plane with a double-napped cone. They include ellipses, parabolas, hyperbolas, and circles. Understanding conic sections is crucial in analytic geometry as they have distinct algebraic representations and geometric properties which are applied in various fields such as physics, engineering, and astronomy.

Ellipse

An ellipse is a type of conic section where the sum of the distances from any point on the curve to two fixed points (the foci) is constant. Its shape is an elongated circle, and it is characterized by a center, two axes (major and minor), and a specific degree of roundness described by its eccentricity.

Standard Form of an Ellipse

The standard form of an ellipse's equation is written to clearly show its center and the lengths of its semi-major and semi-minor axes. This form, typically given as (x - h)²/a² + (y - k)²/b² = 1, directly reveals the geometric features of the ellipse and simplifies further analysis such as graphing and identifying other properties.

Completing the Square

Completing the square is an algebraic technique that involves rewriting a quadratic expression as a perfect square plus a constant. This method is instrumental in transforming the general quadratic equation of a conic section into its standard form, thereby making it easier to identify key features such as the center, axes, and orientation of the curve.

Ellipse Parameters (Semi-Axes, Focal Distance, and Eccentricity)

Ellipse parameters include the semi-major axis (a) and semi-minor axis (b), from which other important quantities are derived. The focal distance (c) is calculated using the relationship c² = a² - b² for ellipses, and the eccentricity (e) is defined as e = c/a. These parameters together characterize the size, shape, and elongation of the ellipse, offering a complete description of its geometric structure.

Transcript

00:03 So let's say that we have this equation, 9x squared plus 4y squared minus 18x plus 16y equals 1.

00:10 And we are asked to write this in the standard form of an ellipse.

00:13 So let's be called the standard form of an ellipse.

00:16 It is x minus h squared over a squared plus y minus k squared over b squared equals 1.

00:32 So basically we need to make sure, we need to condense all of this, or this entire equation to make perfect squares like this.

00:41 So the first thing i want to do is i want to move all of our xs together and all of our ys together.

00:46 So that's just a simple rewrite.

00:48 So we have 9x squared minus 18x plus 4y squared plus 16y equals 1.

01:01 So now i notice that there are some factors.

01:03 Some factors that they share.

01:07 There's a 9 here and an 18 here, a 4 here, and a 16 here.

01:11 So we can pull out some common factors.

01:14 So let's pull out our 9.

01:16 So we have 9 times x squared minus 2x plus 4 times y squared plus 4 y equals 1.

01:29 Now in order to get these perfect squares here, we must complete the square.

01:34 So in order to complete the square, we have to add a constant to this binomial to make it a trinomial or perfect squares trinomial.

01:45 If you recall the format of a perfect square's trinomial, it's x squared minus, let's say, a x plus b squared.

01:55 And b squared is simply a constant...

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