sketch the curve represented by the parametric equations. Then eliminate and write the corresponding rectangular equation whose graph represents the curve.
x=(√t2+2)
y=t/4
square both equations
x^2=t^2+2 and
16y^2=t^2
x^2=16y^2+2
x^2-(4y)^2=2
To sketch the curve represented by the parametric equations, we can proceed as follows:
Step 1: Determine the range of the parameter t.
Since there are no specific restrictions mentioned, we can assume that t can take any real value.
Step 2: Select some values of t and evaluate the corresponding x and y coordinates.
Let's choose a few values of t and calculate x and y coordinates:
For t = 0:
x = √(0^2 + 2) = √2 ≈ 1.414
y = 0/4 = 0
For t = 1:
x = √(1^2 + 2) = √3 ≈ 1.732
y = 1/4 = 0.25
For t = 2:
x = √(2^2 + 2) = √6 ≈ 2.449
y = 2/4 = 0.5
For t = -1:
x = √((-1)^2 + 2) = √3 ≈ 1.732
y = -1/4 ≈ -0.25
For t = -2:
x = √((-2)^2 + 2) = √6 ≈ 2.449
y = -2/4 = -0.5
Step 3: Plot the obtained points.
Plot the points (x, y) from Step 2 on an xy-plane.
Step 4: Connect the plotted points.
Draw a smooth curve that passes through the plotted points to represent the sketch of the curve.
To eliminate the parameter t and write the corresponding rectangular equation, we can express t in terms of x and substitute it into the equation for y:
Given:
x = √(t^2 + 2)
y = t/4
Squaring both sides of the equation for x:
x^2 = t^2 + 2
Rearranging the equation for t:
t = √(x^2 - 2)
Substituting the expression for t into the equation for y:
y = √(x^2 - 2)/4
Therefore, the rectangular equation whose graph represents the curve is y = √(x^2 - 2)/4.