Problem 10 - Thomas Calculus

Question

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(3 x^{2}+6 x y+3 y^{2}-4 x+5 y=12\)

Step-by-Step Solution

Answer

The given equation represents a parabola.

Step 1: Rewrite the Equation

Start by rewriting the given equation in the general form of a conic section: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] Given: \[ 3x^2 + 6xy + 3y^2 - 4x + 5y - 12 = 0 \]Here, \( A = 3 \), \( B = 6 \), \( C = 3 \), \( D = -4 \), \( E = 5 \), and \( F = -12 \).

Step 2: Identify the Coefficients

From the equation, identify the coefficients: \( A = 3 \), \( B = 6 \), and \( C = 3 \). These are needed to calculate the discriminant.

Step 3: Calculate the Discriminant

The discriminant of a conic section is given by the formula:\[ B^2 - 4AC \]Substitute the values of \( A \), \( B \), and \( C \) into the formula:\[ (6)^2 - 4(3)(3) = 36 - 36 = 0 \]

Step 4: Analyze the Discriminant

The value of the discriminant is 0. This suggests that the conic section represents a parabola, since for parabolas the discriminant \( B^2 - 4AC \) equals zero.

Key Concepts

Discriminant

The discriminant is a crucial algebraic tool to classify conic sections such as parabolas, ellipses, and hyperbolas. For any general quadratic equation in two variables of the form:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]you can determine the nature of the conic section using the discriminant formula:\[ B^2 - 4AC \]The value of the discriminant can tell you which conic section the equation represents:

• If the discriminant is less than zero (\[ B^2 - 4AC < 0 \]), the equation represents an ellipse.
• If the discriminant equals zero (\[ B^2 - 4AC = 0 \]), the equation forms a parabola.
• If the discriminant is greater than zero (\[ B^2 - 4AC > 0 \]), the equation correlates to a hyperbola.

By looking solely at this value, you can grasp a quick perception into the specific curve without needing to graph it. This becomes especially useful when dealing with more complex equations involving multiple variables.

Parabolas

A parabola is one of the basic shapes in the family of conic sections. This shape is distinct for its unique curve which can open either upwards or downwards, and sometimes sideways, depending on its equation. Parabolas have a directrix and a focus, which are defining components of their geometrical properties.With a standard quadratic equation, if the discriminant is zero (\[ B^2 - 4AC = 0 \]), we can confirm that it is indeed a parabola. The equation then typically takes the form of either \[ y = ax^2 + bx + c \] or its horizontal counterpart \[ x = ay^2 + by + c \].Real-world examples include:

• The shape of satellite dishes
• The trajectory of a ball thrown in the air
• Architectural structures like parabolic arches

Understanding the properties of parabolas can therefore be highly effective in both theoretical and practical applications.

Ellipses

An ellipse is another fascinating shape within conic sections. It looks like an elongated circle and arises when the discriminant of the quadratic equation is negative (\[ B^2 - 4AC < 0 \]). Ellipses have two focal points and reflect such properties in their perimeter calculations, where the sum of distances from any point on the ellipse to the two foci is constant.Ellipses can be seen in astronomical orbits, such as planets around the sun. This linkage to nature emphasizes the real-world importance and frequent appearance of ellipses:

• Whispering galleries, where sound follows the elliptical shape
• The paths of planets and celestial objects
• Designs in oval tracks and racing circuits

Studying ellipses can offer insights into various natural phenomena, and even help engineers and designers create more efficient and functional structures.

Hyperbolas

Hyperbolas are the dramatic siblings of ellipses in the conic sections family. If the discriminant of the quadratic equation is positive (\[ B^2 - 4AC > 0 \]), the curve formed is a hyperbola. A hyperbola consists of two curves that mirror each other, with each arm opening in opposing directions.The geometric properties arise from two foci, just like ellipses, but the nature of their relationship is one of subtraction rather than addition. Key practical examples of hyperbolas include:

• The branches of certain types of antennas
• Several scientific models in physics
• Sonic booms from aircraft as hyperbolic wave fronts

As a student, understanding hyperbolas can enhance your knowledge of various scientific and mathematical principles that utilize this fascinating shape.