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Chapter 11: Problem 15

Give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$x=-3 y^{2}$$

Short Answer

Expert verified
The focus is \((-\frac{1}{12}, 0)\) and the directrix is \(x = \frac{1}{12}\).

Step by step solution

01

Understand the Standard Form of Parabola Equation

A parabola with a horizontal axis has the standard form \( x = ay^2 + by + c \). In this case, the given equation is \( x = -3y^2 \), which can be written as \( x = -3(y^2 + 0y + 0) \).
02

Identify the Coefficient and Opening Direction

Here, the coefficient \(-3\) reveals the parabola's axis is horizontal and it opens to the left, as the coefficient is negative.
03

Calculate the Focus and Directrix

For the standard form \( x = a(y-k)^2 + h \), the focus is \((h + \frac{1}{4a}, k)\) and the directrix is \(x = h - \frac{1}{4a}\). For \(x = -3y^2\), \(a = -3\), \(h=0\), and \(k=0\). Thus, the focus is \(\left(0 + \frac{1}{-12}, 0\right) = \left(-\frac{1}{12}, 0\right)\) and the directrix is \(x = 0 - \frac{1}{-12} = \frac{1}{12}\).
04

Sketch the Parabola

Draw a horizontal parabola opening to the left. Mark the focus at \(\left(-\frac{1}{12}, 0\right)\) and the directrix as the vertical line \(x = \frac{1}{12}\). Label these points on your graph for clarity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Focus of a Parabola
The focus of a parabola plays a crucial role in understanding this unique conic section. It is a specific point inside the parabola used to define the curve's properties. Imagine a spotlight shining from the focus; the parabola would reflect all its light into a parallel beam. In a mathematical context:
  • The focus is equidistant from any point on the parabola and a corresponding point on the directrix.
  • In the equation's standard form for a parabola with a horizontal axis, the focus can be located using the formula \((h + \frac{1}{4a}, k)\).
For example, given the equation \(x = -3y^2\), the focus is \((-\frac{1}{12}, 0)\). This critical point helps in sketching and understanding the parabola's overall shape.
Directrix of a Parabola
The directrix of a parabola is a line that serves as a reference for constructing and understanding the parabola. Together with the focus, it helps define the parabola's set of points.A few characteristics:
  • The directrix is always perpendicular to the axis of symmetry of the parabola.
  • It is located symmetrically opposite to the focus about the vertex.
  • For parabolas oriented horizontally, the directrix is a vertical line given by the formula \(x = h - \frac{1}{4a}\).
In our example where \(x = -3y^2\), the directrix is the vertical line \(x = \frac{1}{12}\). Knowing the directrix allows us to better visualize and sketch the parabola with accuracy.
Horizontal Axis Parabola
A horizontal axis parabola is a special type of parabola that opens sideways. Unlike its vertical counterpart, the arms of a horizontal parabola extend left or right. Characteristics include:
  • The standard form of this parabola's equation is \(x = a(y-k)^2 + h\).
  • The direction of opening (left or right) depends on the sign of the coefficient \(a\).
  • If \(a\) is positive, it opens to the right; if negative, to the left.
For the equation \(x = -3y^2\), the negative coefficient indicates the parabola opens to the left. Recognizing the axis orientation and direction helps in sketching and analyzing the parabola.
Conic Sections
Conic sections are a group of curves formed by intersecting a plane with a cone. These include circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties and equations. Parabolas are distinct in conic sections because:
  • They have one focus and no center.
  • They are defined by a single quadratic expression.
  • Their unique property is that any point on the parabola is equidistant from the focus and the directrix.
Understanding the role of parabolas within conic sections broadens your comprehension of geometry, showcasing the relationships between shapes formed by different intersecting tasks.

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Most popular questions from this chapter

Replace the Cartesian equations with equivalent polar equations. $$(x+2)^{2}+(y-5)^{2}=16$$

Sketch the lines and find Cartesian equations for them. $$r \cos \left(\theta+\frac{\pi}{3}\right)=2$$

Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=1 / 3, \quad y=6$$

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r=2 \cos \theta-\sin \theta$$

Replace the Cartesian equations with equivalent polar equations. $$x^{2}+(y-2)^{2}=4$$
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