Question

For the following equations, determine which of the conic sections is described. $$ 52 x^{2}-72 x y+73 y^{2}+40 x+30 y-75=0 $$

Short answer

Ellipse

Step-by-step solution

Identify the General Equation of Conic Sections

Conic sections are generally expressed in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Compare the given equation with this general form:\[ 52x^2 - 72xy + 73y^2 + 40x + 30y - 75 = 0 \]From this, we identify: \(A = 52\), \(B = -72\), \(C = 73\), \(D = 40\), \(E = 30\), \(F = -75\).

Calculate the Discriminant

The discriminant of a conic section is given by \(B^2 - 4AC\). By substituting the values from the equation:\[ B^2 - 4AC = (-72)^2 - 4 \times 52 \times 73 \]Calculate it to determine the nature of the conic section.\[ B^2 - 4AC = 5184 - 15184 = -10000 \]

Analyze the Discriminant Result

If the discriminant \(B^2 - 4AC\) is:- Less than 0, the conic is an ellipse (possibly a circle, if \(A = C\)).- Equal to 0, the conic is a parabola.- Greater than 0, the conic is a hyperbola.Since \(B^2 - 4AC = -10000 < 0\), the conic section is an ellipse.

Key concepts

Ellipse

An ellipse is a type of conic section that you encounter in both mathematics and real life, often represented as an elongated circle. To understand its properties, imagine two fixed points, called foci, and consider that the sum of the distances from any point on the ellipse to these foci is constant. This gives the ellipse its distinctive, oval shape.

Ellipses have some special characteristics:

• They have a major axis, which is the longest diameter of the ellipse, and a minor axis, the shortest.
• The center of the ellipse is the midpoint of both axes.
• Ellipses are symmetrical about their major and minor axes.

In the world of planets and satellites, orbits are often elliptical due to gravitational forces, making the study of ellipses crucial in astronomy.
Understanding these properties helps recognize an ellipse through its general equation and discerning features in mathematical problems.

Discriminant

The discriminant in the context of conic sections is a valuable tool used to identify the type of conic you are dealing with. It is calculated using the formula \( B^2 - 4AC \), derived from the coefficients of the general conic section equation.

The value of the discriminant tells you:

• If \( B^2 - 4AC < 0 \), you're working with an ellipse. If \( A = C \), it may even be a circle.
• If it equals 0, the equation represents a parabola.
• If \( B^2 - 4AC > 0 \), the conic section is a hyperbola.

This calculation simplifies the process of classifying conic sections without sketching their graph. It provides a straightforward method not only for academic exercises but also in applications where quick identification is essential.

General Equation of Conic Sections

All conic sections may be represented by a general equation, given as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Each type of conic section can be identified by the specific values of the coefficients \( A \), \( B \), and \( C \).

Here's how they guide us:

• If \( A \) and \( C \) are present but \( B = 0 \), you generally have an ellipse or a circle if \( A = C \).
• If \( B eq 0 \), the rotation of axes might give you an ellipse or hyperbola.
• The sum \( B^2 - 4AC \) further clarifies the conic's identity, using the discriminant as we explored above.

Understanding how these coefficients interact within the equation provides insights into the geometric nature of graphical representations. Thus, linking algebraic expressions to their geometric counterparts makes the concept of conic sections more intuitive and applicable.