Problem 3
Question
The graph of the equation \(y^{2}=4 p x\) is a parabola with focus \(F\) (____, ____) and directrix \(x=\) ____. So the graph of \(y^{2}=12 x\) is a parabola with focus \(F\) (____, ____) and directrix x= ____.
Step-by-Step Solution
Verified Answer
Focus is (3, 0) and directrix is x = -3.
1Step 1: Identify the Parabola Form
The given equation is in the form of a parabola, which is generally written as \( y^2 = 4px \), where \( p \) represents the distance from the vertex to the focus as well as from the vertex to the directrix.
2Step 2: Extract the Parabola's Parameters
For the equation \( y^2 = 12x \), compare it with the standard form \( y^2 = 4px \). Here, \( 4p = 12 \).
3Step 3: Solve for p
Simplify the equation \( 4p = 12 \) to find \( p \). Divide both sides by 4: \( p = 3 \).
4Step 4: Determine the Focus and Directrix
For a parabola \( y^2 = 4px \), the focus is located at \( (p, 0) \). Since \( p = 3 \), the focus will be at \( (3, 0) \). The directrix is the line \( x = -p \), which is \( x = -3 \) when \( p = 3 \).
Key Concepts
Focus of a parabolaDirectrix of a parabolaStandard form of a parabola
Focus of a parabola
The focus of a parabola is a specific point that is situated along the axis of symmetry of the parabola. It has the unique property that any point on the parabola is equidistant from this focus and a corresponding line known as the directrix.
In the context of the standard form of the parabola equation, specifically when you see an equation like \( y^2 = 4px \), the focus is located at \((p, 0)\).
What this means practically is that the distance \( p \) from the vertex to the focus is a crucial figure in determining the shape and direction of the parabola.
For example, when we have the equation \( y^2 = 12x \), solving \( 4p = 12 \) gives us \( p = 3 \).
This positions the focus at the point \( (3, 0) \) on the Cartesian plane.
Recognizing and correctly identifying the focus helps in accurately sketching the parabola and understanding its geometric properties.
In the context of the standard form of the parabola equation, specifically when you see an equation like \( y^2 = 4px \), the focus is located at \((p, 0)\).
What this means practically is that the distance \( p \) from the vertex to the focus is a crucial figure in determining the shape and direction of the parabola.
For example, when we have the equation \( y^2 = 12x \), solving \( 4p = 12 \) gives us \( p = 3 \).
This positions the focus at the point \( (3, 0) \) on the Cartesian plane.
Recognizing and correctly identifying the focus helps in accurately sketching the parabola and understanding its geometric properties.
Directrix of a parabola
The directrix of a parabola acts as a guiding line that ensures the parabolic shape maintains its symmetric properties. Parabolas have an equal distance relationship between points on the curve, the focus, and the directrix.
For a parabola represented by the equation \( y^2 = 4px \), the directrix is defined by the line \( x = -p \).
This means that each point on the parabola is equidistant from both the focus \((p, 0)\) and this line \( x = -p \).
In our specific parabolic equation \( y^2 = 12x \), identifying \( p \, \) as \( 3 \), the directrix becomes \( x = -3 \).
This line serves as an invisible boundary that maintains the parabola's consistent orientation regardless of how large or how small the parabola happens to be.
For a parabola represented by the equation \( y^2 = 4px \), the directrix is defined by the line \( x = -p \).
This means that each point on the parabola is equidistant from both the focus \((p, 0)\) and this line \( x = -p \).
In our specific parabolic equation \( y^2 = 12x \), identifying \( p \, \) as \( 3 \), the directrix becomes \( x = -3 \).
This line serves as an invisible boundary that maintains the parabola's consistent orientation regardless of how large or how small the parabola happens to be.
Standard form of a parabola
Understanding the standard form of a parabola helps in grasping the fundamental attributes of the curve and in solving many practical problems.
The standard form of a parabola when it opens along the x-axis is \( y^2 = 4px \) where \( p \) represents the distance between the vertex and the focus or the vertex and the directrix.
This form is essential because it simplifies the process of identifying key features of the parabola such as its focus and directrix straight from the equation.
In practice, to transform an equation into this standard form, you can compare it to \( y^2 = 4px \) and solve for \( p \).
For instance, in the equation \( y^2 = 12x \), you identify \( 4p = 12 \), giving \( p = 3 \).
Knowing \( p \), you can effortlessly deduce further details about the parabola, making this form incredibly valuable for detailed graph analysis and applications in geometry.
The standard form of a parabola when it opens along the x-axis is \( y^2 = 4px \) where \( p \) represents the distance between the vertex and the focus or the vertex and the directrix.
This form is essential because it simplifies the process of identifying key features of the parabola such as its focus and directrix straight from the equation.
In practice, to transform an equation into this standard form, you can compare it to \( y^2 = 4px \) and solve for \( p \).
For instance, in the equation \( y^2 = 12x \), you identify \( 4p = 12 \), giving \( p = 3 \).
Knowing \( p \), you can effortlessly deduce further details about the parabola, making this form incredibly valuable for detailed graph analysis and applications in geometry.
Other exercises in this chapter
Problem 3
The graph of the equation \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) with \(a>b>0\) is an ellipse with vertices (________ , _______) and (_______ , _______)
View solution Problem 3
The graphs of \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\) and \(\frac{(x-3)^{2}}{5^{2}}+\frac{(y-1)^{2}}{4^{2}}=1\) are given. Label the vertices and foci on
View solution Problem 3
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity $$\$fTrueac{2}{3},$$ directrix $x=3
View solution Problem 4
Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(-2,1), \quad \phi=30^{\circ}$$
View solution