Question

How do I find the standard equation of a parabola given 3 points?

Answer 1

Hint: In this question, we need to find the method of finding the standard equation of a parabola if we are given any three points. For this, we will take general equation of parabola $ y=a{{x}^{2}}+bx+c $ . Putting each point in the equation will give us the three conditions for three variables. Solving them will give us the value of a, b, c, and then the equation of a parabola. We will also understand the method with the help of an example.

Complete step by step answer:
Here we need to find the method of finding the standard equation of a parabola if we are given any three points. Let us suppose that, these three points are $ \left( {{x}_{1}},{{y}_{1}} \right),\left( {{x}_{2}},{{y}_{2}} \right),\left( {{x}_{3}},{{y}_{3}} \right) $ . We know that, the general equation of the parabola is given by $ y=a{{x}^{2}}+bx+c $ . So this is an equation with three unknown variables. Finding them will give us the equation of the parabola. So if we put each point into the equation we can get three equation and solving them will give us value of a, b and c.
We know that if a point $ \left( {{x}_{0}},{{y}_{0}} \right) $ lies on curve $ y=a{{x}^{2}}+bx+c $ then it satisfies the curve i.e. $ {{y}_{0}}=a{{x}_{0}}^{2}+b{{x}_{0}}+c $ .
So since we are given three points and we supposed them as $ \left( {{x}_{1}},{{y}_{1}} \right),\left( {{x}_{2}},{{y}_{2}} \right),\left( {{x}_{3}},{{y}_{3}} \right) $ . So these will satisfy the equation. Thus,
Putting $ \left( {{x}_{1}},{{y}_{1}} \right) $ in $ y=a{{x}^{2}}+bx+c $ we get $ {{y}_{1}}=a{{x}_{1}}^{2}+b{{x}_{1}}+c $ .
Rearranging we get $ {{y}_{1}}-a{{x}_{1}}^{2}+b{{x}_{1}}-c=0\cdots \cdots \cdots \left( 1 \right) $ .
Putting $ \left( {{x}_{2}},{{y}_{2}} \right) $ in $ y=a{{x}^{2}}+bx+c $ we get $ {{y}_{2}}=a{{x}_{2}}^{2}+b{{x}_{2}}+c $ .
Rearranging we get $ {{y}_{2}}-a{{x}_{2}}^{2}+b{{x}_{2}}-c=0\cdots \cdots \cdots \left( 2 \right) $ .
Putting $ \left( {{x}_{3}},{{y}_{3}} \right) $ in $ y=a{{x}^{2}}+bx+c $ we get $ {{y}_{3}}=a{{x}_{3}}^{2}+b{{x}_{3}}+c $ .
Rearranging we get $ {{y}_{3}}-a{{x}_{3}}^{2}-b{{x}_{3}}-c=0\cdots \cdots \cdots \left( 3 \right) $ .
Therefore solving (1), (2), and (3) will give us the value of a, b and c. Putting them in the general equation of parabola we get our required equation.
Let us understand with an example where we are given three points as (0,0), (1,1) and (-1,1)
Standard equation is $ y=a{{x}^{2}}+bx+c $ .
Putting (x,y) as (0,0) we get $ 0=a{{\left( 0 \right)}^{2}}+b\left( 0 \right)+c\Rightarrow c=0 $ .
Putting (x,y) as (1,1) we get $ 1=a{{\left( 1 \right)}^{2}}+b\left( 1 \right)+c\Rightarrow a+b+c=1 $ .
Putting c = 0 we get $ a+b=1\Rightarrow a=1-b $ .
Putting (x,y) as (-1,1) we get $ 1=a{{\left( -1 \right)}^{2}}+b\left( -1 \right)+c\Rightarrow 1=a-b+c $ .
Putting c = 0 and value of a as 1-b we get,
 $ 1=1-b-b+0\Rightarrow 1=1-2b\Rightarrow 2b=0\Rightarrow b=0 $ .
Therefore, a = 1-0 = 1.
So value of a = 1, b = 0, c = 0.
Putting them in $ y=a{{x}^{2}}+bx+c $ we get $ y={{x}^{2}} $ which is the required equation of parabola.

Note:
Students should note that, when we obtain three equation, there is a chance that we do not obtain any solution. Students must know how to solve three equations with three unknown variables.