Find the Cartesian equation of the curve

In summary, the conversation discusses finding the solution for the equations ##xt=t^2+2## and ##yt=t^2-2##, and using basic identities to show that the curve is a hyperbola and that ##x^2-y^2=8##.
  • #1

chwala

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Homework Statement
This is a past paper question...see attached (my interest is on the highlighted) I used a different approach thus your insight would be great. I always make it a point to try out the questions before checking what the mark scheme offers...
Relevant Equations
parametric equations.
1671348574228.png

Find ms solution;

1671348656844.png


My approach;

##xt=t^2+2## and ##yt=t^2-2##

##xt-2=t^2## and ##yt+2=t^2##

##⇒xt-2=yt+2##

##xt-yt=4##

##t(x-y)=4##

##t=\dfrac{4}{x-y}##

We know that;

##x+y=2t##

##x+y=2⋅\dfrac{4}{x-y}##

##(x-y)(x+y)=8##

##x^2-y^2=8##

Your insight is welcome...rather asking if this approach would be correct.
 
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  • #2
[tex]x+y=2t[/tex]
[tex]x-y=\frac{4}{t}[/tex]
[tex](x+y)(x-y)=8[/tex]
 
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Likes pasmith, topsquark, SammyS and 1 other person
  • #3
Writing [tex]
\begin{split}
x &= t + \frac{2}{t} = 2\sqrt{2}\operatorname{sgn}(t)\cosh(\ln(|t|/\sqrt{2})) \\
y &= t - \frac{2}{t} = 2\sqrt{2}\operatorname{sgn}(t)\sinh(\ln(|t|/\sqrt{2}))\end{split}[/tex] shows that the curve is a hyperbola, and basic identities then give [tex]x^2 - y^2 = (2\sqrt{2})^2 = 8.[/tex]
 
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1. What is a Cartesian equation?

A Cartesian equation is an algebraic equation that describes the relationship between the x and y coordinates of points on a two-dimensional plane. It is named after the French philosopher and mathematician René Descartes, who developed the Cartesian coordinate system.

2. How is a Cartesian equation different from other types of equations?

A Cartesian equation is unique in that it uses the x and y coordinates to represent points on a two-dimensional plane. This allows for a visual representation of the equation through a graph, making it easier to understand and analyze.

3. How do I find the Cartesian equation of a curve?

To find the Cartesian equation of a curve, you need to know the coordinates of at least three points on the curve. Then, you can use the slope formula to find the slope of the line between each pair of points. Finally, you can use the point-slope form of a line to write the equation in the form y = mx + b, where m is the slope and b is the y-intercept.

4. Can a Cartesian equation represent any type of curve?

Yes, a Cartesian equation can represent any type of curve, including straight lines, parabolas, circles, and more complex curves. The shape of the curve will depend on the specific values and variables in the equation.

5. How can I use a Cartesian equation in real life?

Cartesian equations have many practical applications in fields such as physics, engineering, and economics. They can be used to model and analyze various phenomena, such as the motion of objects, the growth of populations, and the relationships between variables in a system.

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