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Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
Use the Distance Formula to find the y-coordinate of B. If necessary, round your answer to 2 decimal places.
Is it possible to find a point P directly above F so that the distance from P to F and the distance from P to d are the same?
The equation of a parabola can be found using the Distance Formula.
Consider the previous parabola, this time drawn on a coordinate plane. The focus of the parabola is F(0,2), and its directrix is the line with the equation y=-2. Consider also a point P with an x-coordinate of 3, lying on the parabola.
What is the y-coordinate of P?Use the Distance Formula to express the distance from the focus F(0,2) to the point P(3,y).
Substitute values
Subtract term
LHS2=RHS2
(a±b)2=a2±2ab+b2
LHS−y2=RHS−y2
LHS−4=RHS−4
LHS+4y=RHS+4y
LHS/8=RHS/8
ba=b1⋅a
Keeping the previous example in mind, consider a parabola with focus (0,p) and directrix y=-p. How can its equation be obtained?
By definition, any point P(x,y) on the parabola must be equidistant from the focus and the directrix. This means that FP and RP are congruent segments. Therefore, they have the same length. The Distance Formula can be used to write an expression for each length.
d=(x2−x1)2+(y2−y1)2 | ||
---|---|---|
Points | Substitution | Simplififcation |
F(0,p) and P(x,y) | FP = (x−0)2+(y−p)2 | FP = x2+(y−p)2 |
R(x,-p) and P(x,y) | RP = (x−x)2+(y−(-p))2 | RP = (y+p)2 |
LHS2=RHS2
(a±b)2=a2±2ab+b2
LHS−y2=RHS−y2
LHS−p2=RHS−p2
LHS+2yp=RHS+2yp
LHS/4p=RHS/4p
ba=b1⋅a
Rearrange equation
Up to this point, parabolas whose directrices are parallel to the x-axis have been discussed. Next, parabolas whose directrices are parallel to the y-axis will be examined.
Izabella is making an original video game character. She wants a force field in the shape of a parabola. When the character is at F(-4,-2) and the opposing team's army is on vertical line x=2, the force field will appear as shown in the graph.
d=(x2−x1)2+(y2−y1)2 | ||
---|---|---|
Points | Substitution | Simplififcation |
F(-4,-2) and P(x,y) | FP = (x−(-4))2+(y−(-2))2 | FP = (x+4)2+(y+2)2 |
R(2,y) and P(x,y) | RP = (x−2)2+(y−y)2 | RP = (x−2)2 |
LHS2=RHS2
(a±b)2=a2±2ab+b2
LHS−x2=RHS−x2
LHS+4x=RHS+4x
LHS−16=RHS−16
LHS−(y+2)2=RHS−(y+2)2
LHS/12=RHS/12
Write as a difference of fractions
Put minus sign in front of fraction
ba=b1⋅a
aa=1