Using parametric equations in coordinate geometry
This article is a continuation from the previous article. This article covers how to use parametric equations in coordinate geometry. You can use parametric equations to solve problems in coordinate geometry. We can see how this is done in the following examples.
Example
In this example we’re going to find a set of coordinates using a curve and the x-axis. Suppose we had to find the coordinates where the curve with the following parametric equations meet the x-axis.
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Below is the curve with the set of coordinates we’re trying to find labelled A and B.
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On the x-axis y is always zero so we don’t need to worry about y. We only need to find the value of x. We know that;
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We can use y=4-t² to find the value of t that we can use in the equation above. Notice that we can find the value of t since we know y=0 at the x-axis.
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…therefore when t=±2.
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…when x = -2;
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We have managed to identify the coordinates as being;
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Example
In this example we’re going to find the coefficients of the parametric equations given a particular coordinate. The parametric equations below define a curve;
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…where a is a constant. We’re also told that the curve passes through (2, 0). Suppose we had to find the value of a.
This time we know the value of y and x which is universal to the parametric equations. So we can use the known value of y in the equation y=(t³ + 8) to find the value of t then use t in x=at to find the value of t.
When y=0.
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The cube root of -8 is -2.
Now we can use t in the equation x=at to find the value of a;
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The value of a=-1. So the parametric equations of the curve are x=-t and y=-(t³ + 8).
Example
In this example we’re going to find the coordinates where the following line and the curve given by the parametric equations meet;
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To solve the problem we first solve the equations simultaneously to find the value of t. We can substitute the parametric equations into the line equation to find the value of t as shown below;
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Now we can substitute t in the parametric equations to find the values of y and x for the coordinates; for x=t²
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So x=4, and for y=4t;
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So the value of y=-8 that must mean coordinates where the curve and the line meet are;
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