Conic Section Analysis: Ellipse, Foci, Center, And Eccentricity

Let's dive deep into the world of conic sections! In this article, we'll dissect the given conic equation, 4x² - 8x + 9y² - 18y – 23 = 0, and meticulously analyze its properties. We'll determine its type, pinpoint its foci and center, and calculate its eccentricity. So, buckle up, guys, it's going to be a mathematically thrilling ride!

Evaluating the Conic Section

Our starting point is the equation 4x² - 8x + 9y² - 18y – 23 = 0. The first step in understanding this conic is to rewrite it in its standard form. This involves completing the square for both the x and y terms. This process will unveil the true nature of the conic section, allowing us to easily identify its key parameters. By converting to standard form, we're essentially revealing the conic's hidden structure, much like an architect's blueprint reveals the layout of a building.

Completing the Square

Let's start by grouping the x and y terms together: (4x² - 8x) + (9y² - 18y) – 23 = 0. Next, we factor out the coefficients of the squared terms: 4(x² - 2x) + 9(y² - 2y) – 23 = 0. Now comes the magic of completing the square. Remember, to complete the square for an expression like x² + bx, we add (b/2)² to it. For the x terms, (b/2)² = (-2/2)² = 1. For the y terms, (b/2)² = (-2/2)² = 1 as well.

We add and subtract these values inside the parentheses, being careful to account for the factors we pulled out earlier: 4(x² - 2x + 1 - 1) + 9(y² - 2y + 1 - 1) – 23 = 0. This can be rewritten as 4((x - 1)² - 1) + 9((y - 1)² - 1) – 23 = 0. Expanding the terms gives us 4(x - 1)² - 4 + 9(y - 1)² - 9 – 23 = 0. Finally, we move the constants to the right side: 4(x - 1)² + 9(y - 1)² = 36.

To get the standard form, we divide both sides by 36: (x - 1)²/9 + (y - 1)²/4 = 1. Aha! This is the standard form of an ellipse. We've successfully unveiled the conic's identity through the algebraic manipulation of completing the square. This standard form is our key to unlocking all the secrets of this ellipse.

Identifying the Conic

From the standard form (x - 1)²/9 + (y - 1)²/4 = 1, we can immediately identify this conic as an ellipse. The presence of a '+' sign between the squared terms and different denominators under the x² and y² terms are telltale signs of an ellipse. If the denominators were equal, it would be a circle (a special case of an ellipse). If the sign between the terms was '-', it would be a hyperbola. And if only one variable was squared, it would be a parabola.

Now that we've confirmed it's an ellipse, we can extract some crucial information. The larger denominator, 9, is under the (x - 1)² term, indicating that the major axis is horizontal. The square root of 9, which is 3, represents the semi-major axis (a). The square root of 4, which is 2, represents the semi-minor axis (b). These values, a and b, are fundamental to understanding the ellipse's shape and dimensions. We're building a complete picture of our conic section, piece by piece.

Analyzing the Statements

Now, let's tackle the statements provided and see which ones hold true. This involves using the information we've gleaned from the standard form and applying our knowledge of ellipses.

Statement I: It is an ellipse.

We've already established this! Through the process of completing the square and examining the standard form, we definitively concluded that the conic section is an ellipse. So, Statement I is indeed correct. We're off to a good start! This initial verification sets the stage for a more in-depth analysis of the ellipse's characteristics.

Statement II: It has foci at (-4, 0) and (5, 0).

To determine the foci, we need to calculate the distance from the center to each focus, denoted by 'c'. The relationship between a, b, and c in an ellipse is given by the equation c² = a² - b². In our case, a² = 9 and b² = 4, so c² = 9 - 4 = 5. Therefore, c = √5.

The center of the ellipse is at (1, 1), which we can read directly from the standard form: (x - 1)²/9 + (y - 1)²/4 = 1. Since the major axis is horizontal, the foci will lie along the horizontal line passing through the center. To find the foci, we move √5 units to the left and right of the center.

Thus, the foci are located at (1 - √5, 1) and (1 + √5, 1). These are clearly not the points (-4, 0) and (5, 0) stated in Statement II. So, Statement II is incorrect. This highlights the importance of accurate calculations and careful application of formulas when analyzing conic sections. The foci are critical points that define the shape and characteristics of the ellipse.

Statement III: It has a center at (1, 1).

As we mentioned earlier, the center of the ellipse can be directly read from the standard form: (x - 1)²/9 + (y - 1)²/4 = 1. The center is indeed at (1, 1). Therefore, Statement III is correct. Identifying the center is a crucial step in understanding the ellipse's position and orientation in the coordinate plane. It serves as the reference point for determining other key features, such as the foci and vertices.

Statement IV: The eccentricity is a certain value.

The eccentricity (e) of an ellipse is a measure of how elongated it is. It's defined as the ratio of the distance from the center to a focus (c) to the semi-major axis (a): e = c/a. We already calculated c = √5 and identified a = 3. Therefore, the eccentricity is e = √5 / 3.

To fully assess this statement, the