Question

The equation \(2 \mathrm{x}^{2}+3 \mathrm{y}^{2}-8 \mathrm{x}-18 \mathrm{y}+35=\mathrm{k}\) represents (a) parabola if \(\mathrm{k}>0\) (b) circle if \(\mathrm{k}>0\) (c) a point if \(\mathrm{k}=0\) (d) a hyperbola if \(\mathrm{k}>0\)

Short answer

The given equation represents a point if \(k = 0\).

Step-by-step solution

Complete the square for the given equation

First, we will group the terms with x and y together and complete the squares for each group. \[2(x^2 - 4x) + 3(y^2 - 6y) + 35 = k\] To complete the square for the x terms, we need to add and subtract \(\frac{4^2}{4} = 4\), and for the y terms, we need to add and subtract \(\frac{6^2}{4} = 9\): \[2(x^2 - 4x + 4) + 3(y^2 - 6y + 9) - 8 - 27 + 35 = k\] Now, we can rewrite the equation in standard form: \[2(x-2)^2 + 3(y-3)^2 = k + 8 + 27 - 35\] Simplifying the equation, we get: \[2(x-2)^2 + 3(y-3)^2 = k\]

Analyze the standard form

From the standard form of the equation, we can see that \(2(x-2)^2 + 3(y-3)^2 = k\). This equation is of the form of an ellipse (including circle) since both coefficients of the squared terms are positive. Now, we can analyze the conditions for k stated in each option: (a) If k > 0, the equation does not represent a parabola, as a parabola has only one squared term. (b) If k > 0, the equation can represent a circle if the coefficients of the squared terms are equal (which they are not in this case, 2 and 3), so the given equation does not represent a circle. (c) If k = 0, the equation becomes \(2(x-2)^2 + 3(y-3)^2 = 0\), this is only possible if both (x-2)²=0 and (y-3)²=0, which means x=2 and y=3, so this represents a point. (d) If k > 0, the equation does not represent a hyperbola, as a hyperbola has one positive and one negative coefficient for the squared terms. Based on our analysis, the correct answer is: (c) a point if \(k = 0\)

Key concepts

Completing the Square

Completing the square is a pivotal method used in solving quadratic equations, including transformations of conic section equations into their standard forms.

To illustrate completing the square, consider the quadratic equation part of the exercise involving the variable x: \(2x^2 - 8x\). To complete the square, we rewrite this expression to form a perfect square trinomial. We find the necessary constant by taking half of the linear coefficient (in this case, -8 divided by 2), squaring it, and then adding and subtracting this square within the expression. Here it results in \(2(x - 2)^2\), after adding and subtracting 4 inside the parenthesis. A similar process is applied to the y variable, resulting in \(3(y - 3)^2\).

By rearranging terms and simplifying, we're able to transition the equation towards a form that reveals more about the geometric nature of the graph it represents. This technique not only provides us with the vertex of a parabola but also aids in classifying and understanding the central features of other conic sections, such as ellipses and circles.

Conic Sections

Conic sections encompass the various shapes that can be formed by slicing a three-dimensional cone with a plane. These shapes include circles, ellipses, parabolas, and hyperbolas, each with their unique equations and properties.

For instance, a circle and an ellipse both have equations with squared terms of x and y, but for a circle, these coefficients must be equal and positive. A parabola's equation will have only one squared term, signifying its one-directional curvature. A hyperbola, identifying with two oppositely opening arcs, will present with one squared term's coefficient being negative.

The provided exercise involves determining whether the equation corresponds to one of these forms based on the value of k. After completing the square and reformatting the equation, we can exclude the possibility of a parabola or hyperbola, as both squared terms are positive and no squared terms are subtracted. This logical analysis of the coefficients and constants related to the variable terms, guided by our understanding of conic sections, allows us to narrow down the possible shapes represented by the given equation.

Standard Form of an Ellipse

An ellipse, one of the primary conic sections, is governed by a standard form equation which showcases its symmetrical nature about two orthogonal axes. This can be expressed as \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), with (h,k) being the coordinates of the center, and 'a' and 'b' representing the lengths of the semi-major and semi-minor axes, respectively.

In our exercise, after completing the square and rearranging, the resulting equation resembles that of an ellipse's. However, since the coefficients 2 and 3 are not reciprocals of squares of integers, further simplification would be required to express it in the canonical form of an ellipse.

If k were greater than zero, the equation would represent an ellipse, but with k equal to zero, the ellipse collapses to a single point. Thus, only when k equals zero does the equation represent a point, specifically at the coordinates given by setting the squared terms to zero, which play the role of the ellipse's center, (2,3) in this case.