1. ## Parabola(multiple choice)

Find the equation of a parabola w/ axis parallel to the x-axis and passing through
(3/2 , 1),(5,0) and (-1,2)?

i tried solving this problem
first substituting coordinate (5,0) to the general eqn giving me 2 unknowns
substituting the rest of the two coords gives me 4 unknowns each

ending up with 3 unknowns which is impossible to solve
any other way to solve this?

Choices:
A. y^2 -2x -8y -10 = 0
B. y^2 -2x -8y+10 = 0
C. y^2 +2x -8y +10 = 0
D. y^2 -2x +8y +10 = 0

2. ## Re: Parabola(multiple choice)

Hello, TechnicianEngineer!

Find the equation of a parabola w/ axis parallel to the x-axis
and passing through (3/2 , 1),(5,0) and (-1,2).

. . $\displaystyle \begin{array}{ccccccccc}(A) & y^2 -2x -8y -10 \:=\: 0 && (B) & y^2 -2x -8y+10 \:=\:0 \\ \\ (C) & y^2 +2x -8y +10 :=\: 0 && (D) & y^2 -2x +8y +10 \:=\: 0 \end{array}$

Since the axis is parallel to the x-axis, we have a "horizontal" parabola.
. . The general form is: .$\displaystyle x \:=\:ay^2 + by + c$

Substitute the three points:

. . $\displaystyle \begin{array}{cccccccccc}(\frac{3}{2},\,1): & \frac{3}{2} &=& a(1^2) + b(1) + c &\Rightarrow& a + b + c \:=\:\frac{3}{2} \\ \\[-3mm](5,\,0): & 5 &=& a(0^2) + b(0) + c &\Rightarrow& c\:=\:5 \\ \\[-3mm] (\text{-}1,\,2): & \text{-}1 &=& a(2^2) + b(2) + c &\Rightarrow& 4a + 2b + c \:=\:\text{-}1 \end{array}$

Solve the system of equations: .$\displaystyle a = \tfrac{1}{2},\;b = \text{-}\:\!4,\;c = 5$

We have: .$\displaystyle x \:=\:\tfrac{1}{2}y^2 - 4y + 5 \quad\Rightarrow\quad 2x \:=\:y^2 - 8y + 10$

Therefore: .$\displaystyle y^2 - 2x - 8y + 10 \:=\:0\;\;\text{ Answer (B)}$

3. ## Re: Parabola(multiple choice)

thanks soroban

just a little clarification since the general eqn of parabola is Cy^2 + Dx + Ey + F = 0 ,
why does "x" variable not have an unknown constant coefficient Dx and just let D = 1

4. ## Re: Parabola(multiple choice)

Originally Posted by TechnicianEngineer
thanks soroban

just a little clarification since the general eqn of parabola is Cy^2 + Dx + Ey + F = 0 ,
why does "x" variable not have an unknown constant coefficient Dx and just let D = 1
1. To determine the coefficients of the variables you have to use 4 equations, that means you must know 4 points of the parabola.

2. $\displaystyle Cy^2 + Dx + Ey + F = 0~\implies~x=-\frac CD y^2-\frac ED y - \frac FD$
With $\displaystyle a = -\frac CD,\ b= -\frac ED,\ c= - \frac FD$ you'll get Soroban's equation. Then you have

• separated the variables
• you only need 3 points to determine the coefficients.

,