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A Cartesian equation of a curve is finding the single equation of the curve in a standard form where xs and ys are the only variables. Mathematician Rene Descartes invented the concept of Cartesian coordinates or equation in the 17th century which brought a revolution in the world of mathematics by providing first symmetric link between Euclidean geometry. Guidelines provides below will aid to understand the method of solving Cartesian equation in a simple manner.
In mathematics, Cartesian coordinate system is used effectively to understand each point in the plane with two numbers which is called x coordinate and y coordinate of the point. Two perpendicular lines are drawn to define the coordinate named x axis and y axis. An equation-representing locus L in the n-dimensional Euclidean space which has the form
L: f (X1,….,Xn) =0,
Left side of the equation represents some expressions of Cartesian equation x1….xn.
The n-tuples of numbers which is x1…xn are fulfilling the equation are the coordinates of the point of L.
To understand the Cartesian equation, it is essential to understand the conversion of polar equation to Cartesian equation. This process involves five steps which are noted below
As a nuclear physicist, conversion of polar equation into Cartesian equation is necessary to be understood as it is a complex calculation which every student faces challenge with.
In order to understand the conversion, it is essential to understand the form of equation. If the equation incorporates with rs and θs, then it is a polar equation. On the other hand, if the equation incorporates xs and ys, then it is a Cartesian equation. For example, if the equation is 7r=sin (θ), then it is a polar equation.
If the convertible equation is in the polar form and it is to be converted into Cartesian, it should be converted in a way that the equation should be left with xs and ys. On the contrary, if the equation is going to be converted from Cartesian to polar equation, the goal of the equation should be rs and θs. This should be the primary objective to solve the equation successfully. From the above example, to solve the equation successfully, the goal should be to arrive at the equation that only consists of x and y terms.
Some key components should be identified in order to solve the equation successfully. If the equation is at the polar form, one should look for R2 which is equal to the equation X2+Y2. If the equation consists of R con(θ) in the polar form, it should be equal to X in Cartesian form. Similarly, if the equation consist of R sin (θ) in the polar form, it should be equal to Y in Cartesian form.
From the above equation, it is clear that right side of the equation can turn into r sin(θ), however it is missing an r term. The left hand side could turn into 7r2 but it is also missing an r term. Meanwhile both of LHS and RHS misses the same term, both sides should be multiplied by r.
7r= sin (θ)
7r2= r sin (θ)
After understanding step 4, one should start substituting because the equation with terms can be converted easily with substitution.
7 (x2+ y2) = y
The equation can be simplified by combining the like terms. If the equation have x2s and y2s, the square can be completed. A fully simplified equation will be able to evaluate r in terms of θ or y in terms of x. afterward, the terms can be combined and the equation can be simplified. To get exact result, RHS can be simplified to zero if the equation is simplified. Any recurring terms can be also factor out. All three equations are varying degrees of simplification.
5 (x2+y2) = y
5×2+ 5y2-y =0
5×2+y (5y-1)= 0
Therefore, using the above steps, the equation can be converted from polar to Cartesian equation. However, one can find it difficult to eliminate the parameter to find Cartesian equation of the curve. Following should be considered to solve the equation easily.
X= e4t, y= t+9
Primarily, parameters should be eliminated to find the Cartesian equation of the curve. A curve can be sketched and indicated with an arrow the direction in which the curve is traced as the parameter T increases.
x = e4t therefore y is needed to be solved to get rid of the parameter from the y equation.
x= e 4(y-9)
This equation can be simplified further with
Y = t+9
Afterward, natural log should be taken and divided with 4 in both sides.
t should be plugged back into the y equation
y = ln (x)/ 4+9
Both of the above equation are correct and are in terms of Cartesian coordinates. Normally, if the equation is equal to Y is acceptable. The traditional method of equation is given below.
in the above equation, y is isolated variable in the equation hence the set of equation y= ln(x)/ 4+9 is more acceptable and correct than x = e 4(y-9).
Afterwards, as the parameter T increases, the curve should be sketched. Three variables should be used to construct a table t, x and y however this is slightly difficult as there is no limitation on T. there is a exponential function exist therefore X will not be less than 0.
0 1 2 3 4
1 e4 e8 e12 e16
9 10 11 12 13
Here, t is the independent variable and x and y are the dependant variable. Afterwards, a graph should be drawn which should look similar to this.
Following these steps successfully, Cartesian equation can be effortlessly solved and converted from parametric and polar equation. Read More
19 Jan, 2021
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