Determining The Value Of C For A Parabola In A Conic Section Equation

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In the realm of analytical geometry, conic sections hold a place of paramount importance. These curves, born from the intersection of a plane and a double-napped cone, manifest in four distinct forms: circles, ellipses, hyperbolas, and parabolas. Each conic section possesses unique characteristics and equations that govern its shape and position in the coordinate plane. In this article, we delve into the specifics of identifying a parabola from its general conic section equation, focusing on determining the value of the coefficient C that dictates the curve's parabolic nature.

Decoding the General Conic Section Equation

The equation at hand, $2x^2 + Cy^2 - 12x - 10y = 96$, represents a general conic section. The presence of both $x^2$ and $y^2$ terms immediately indicates that we are not dealing with a straight line. The coefficients of these squared terms, namely 2 and C, play a pivotal role in determining the specific type of conic section the equation represents. To decipher whether this equation describes a parabola, we must scrutinize the relationship between these coefficients.

The Discriminant's Tale

In the grand tapestry of conic sections, the discriminant emerges as a crucial tool for classification. For a general second-degree equation of the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, the discriminant is defined as $B^2 - 4AC$. This seemingly simple expression holds the key to unlocking the conic section's identity. The sign of the discriminant dictates the curve's nature:

  • If $B^2 - 4AC < 0$, the equation represents an ellipse (or a circle if A = C).
  • If $B^2 - 4AC > 0$, the equation represents a hyperbola.
  • If $B^2 - 4AC = 0$, the equation represents a parabola.

In our given equation, $2x^2 + Cy^2 - 12x - 10y = 96$, we have A = 2, B = 0 (since there is no xy term), and C = C. Applying the discriminant condition for a parabola, we get:

02βˆ’4(2)(C)=00^2 - 4(2)(C) = 0

This simplifies to:

βˆ’8C=0-8C = 0

Therefore, the value of C must be 0 for the equation to represent a parabola.

The Parabola's Defining Characteristic

Now, let's understand why C = 0 leads to a parabola. A parabola, in its essence, is defined as the set of all points equidistant to a fixed point (the focus) and a fixed line (the directrix). This geometric definition translates into an algebraic condition where only one variable is squared. In our equation, setting C = 0 eliminates the $y^2$ term, leaving us with an equation where only x is squared:

2x2βˆ’12xβˆ’10y=962x^2 - 12x - 10y = 96

This form is characteristic of a parabola that opens either upwards or downwards. If the coefficient of the $y^2$ term were zero, and the coefficient of the $x^2$ term was non-zero, we would have a horizontal parabola. The absence of either the $x^2$ or $y^2$ terms, but not both, is the hallmark of a parabolic equation.

Completing the Square to Reveal the Parabola

To further solidify our understanding, let's complete the square for the x terms in the equation $2x^2 - 12x - 10y = 96$. This process will transform the equation into a standard parabolic form, making its properties readily apparent.

First, divide the equation by 2 to simplify the coefficient of $x^2$:

x2βˆ’6xβˆ’5y=48x^2 - 6x - 5y = 48

Next, move the y term and the constant to the right side of the equation:

x2βˆ’6x=5y+48x^2 - 6x = 5y + 48

Now, complete the square on the left side by adding and subtracting the square of half the coefficient of the x term, which is $(-6/2)^2 = 9$:

x2βˆ’6x+9=5y+48+9x^2 - 6x + 9 = 5y + 48 + 9

This simplifies to:

(xβˆ’3)2=5y+57(x - 3)^2 = 5y + 57

Finally, factor out the 5 on the right side:

(x - 3)^2 = 5(y + rac{57}{5})

This equation is now in the standard form of a parabola that opens upwards: $(x - h)^2 = 4p(y - k)$, where (h, k) is the vertex of the parabola and p is the distance between the vertex and the focus, and between the vertex and the directrix. Our equation confirms that we indeed have a parabola, with a vertex at (3, -57/5) and opening upwards.

The Tangible Impact of C = 0

The significance of C = 0 extends beyond mere algebraic manipulation. It has a profound geometric implication. When C = 0, the equation represents a parabola, a curve with a single axis of symmetry. This axis runs parallel to the y-axis in our case, as the x term is squared. The parabola's shape is characterized by its U-like form, which can open upwards, downwards, leftwards, or rightwards, depending on the coefficients in the equation.

Visualizing the Transformation

Imagine starting with a general conic section equation where both $x^2$ and $y^2$ terms are present (and have coefficients of the same sign). This could represent an ellipse or a circle. As we decrease the coefficient of the $y^2$ term, the conic section stretches along the y-axis, becoming more elongated. When this coefficient reaches zero (C = 0), the curve transforms into a parabola, losing its elliptical or circular form entirely. The parabola marks the boundary between ellipses/circles and hyperbolas in the family of conic sections.

Real-World Manifestations of Parabolas

Parabolas are not merely abstract mathematical constructs; they permeate the real world in myriad ways. Their unique reflective property, where parallel rays of light or radio waves converge at the focus, makes them indispensable in various applications:

  • Satellite Dishes: Satellite dishes are parabolic reflectors that focus incoming radio waves onto a receiver placed at the focus.
  • Telescopes: Reflecting telescopes utilize parabolic mirrors to gather and focus light from distant celestial objects.
  • Flashlights and Headlights: The reflectors in flashlights and headlights are parabolic, directing the light into a concentrated beam.
  • Projectile Motion: The trajectory of a projectile, such as a ball thrown in the air, approximates a parabolic path (ignoring air resistance).
  • Suspension Bridges: The cables of suspension bridges often hang in a parabolic shape, distributing the load evenly.

The Parabola's Enduring Legacy

From the graceful curves of bridges to the precision of optical instruments, parabolas play a vital role in shaping our world. Understanding their properties and equations, particularly the conditions that dictate their formation, is crucial for engineers, scientists, and mathematicians alike. The seemingly simple condition of C = 0 in our equation unlocks a world of parabolic wonders, highlighting the elegance and power of analytical geometry.

Conclusion: The Parabolic Threshold

In conclusion, the equation $2x^2 + Cy^2 - 12x - 10y = 96$ represents a parabola if and only if the value of C is 0. This condition stems from the fundamental definition of a parabola and its discriminant, as well as the algebraic requirement of having only one squared variable. By setting C = 0, we eliminate the $y^2$ term, transforming the equation into a parabolic form. This understanding not only enriches our mathematical knowledge but also provides a gateway to appreciating the diverse applications of parabolas in the world around us. The parabolic threshold of C=0 is more than just a mathematical curiosity; it's a gateway to understanding a ubiquitous and powerful geometric shape.

Final Thoughts: Embracing the Beauty of Conic Sections

Conic sections, with their elegant curves and diverse properties, offer a captivating glimpse into the world of mathematics. From the perfect symmetry of circles to the unbounded nature of hyperbolas, each conic section holds its own unique charm. By mastering the techniques for identifying and analyzing these curves, we unlock a powerful toolkit for solving problems in geometry, physics, and engineering. The journey through conic sections is a testament to the beauty and practicality of mathematical concepts, and the condition for a parabola serves as a key milestone in this intellectual exploration.