The figure below shows a conchoid of Nicomedes. The top part is formed by a ray from the origin that rotates through an angle t° with the
x-axis.
The ray intersects the fixed line y = 2 at point A. From A you measure out 7 more units to point P on the graph of the conchoid. The bottom part is formed when the ray is in Quadrants III and IV and you measure 7 units from where the line containing the ray intersects the fixed line.
The parametric equation for x has two parts, one for the segment from the origin to the point on the x-axis below point A, the other from there to the point on the x-axis below point P. Write an equation for x as a function of t.
x = 2 t + t
Write the parametric equation for y as a function of t. It, too, will have two parts.
y = + t
The Cartesian equation of this conchoid is
(x^2 + y^2)(y - 2)^2 = 49y^2
Verify that this equation is correct by calculating the values of x if y is 8 and showing that the points really are on the conchoid.
(Round to the nearest hundredth.)
Now this is getting out of control. You try.
To find the equation for x as a function of t, we need to consider the two parts separately.
The first part is the segment from the origin to the point on the x-axis below point A. Since the line y = 2 is the x-axis, when the ray intersects it, the y-coordinate of the point of intersection (point A) is 2. So, the x-coordinate of point A is simply 0.
The second part is from point A to the point on the x-axis below point P. Here, we measure out 7 units from point A. As the ray rotates, the distance between the point of intersection with the line y = 2 and the x-axis changes. Let's call this distance d. When we measure 7 units from point A, we reach point P. So, the x-coordinate of point P is basically d.
In general, d can be found using trigonometry. Since we are rotating the ray with an angle t°, the value of d can be found using the equation d = 7 sec(t).
Therefore, the equation for x as a function of t is:
x = 0 + 7 sec(t) = 7 sec(t)
Now, let's find the equation for y as a function of t. Like before, we will consider the two parts separately.
For the first part, the y-coordinate remains constant at y = 2, as the ray rotates from the origin to point A.
For the second part, we measure out a distance of 7 units from the line containing the ray and intersects the line y = 2. Let's call this distance k. Again, using trigonometry, we can find k as k = 7 tan(t).
Therefore, the equation for y as a function of t is:
y = 2 + 7 tan(t)
To verify the given Cartesian equation for the conchoid, substitute the values of x and y into the equation and check if it holds true.
Given: (x^2 + y^2)(y - 2)^2 = 49y^2
Substitute y = 8 into the equation: (x^2 + 8^2)(8 - 2)^2 = 49(8^2)
Simplify: (x^2 + 64)(6^2) = 49(64)
Expand: 36(x^2 + 64) = 49(64)
Distribute: 36x^2 + 2304 = 3136
Solve for x^2: 36x^2 = 3136 - 2304
Simplify: 36x^2 = 832
Divide by 36: x^2 = 23.11
Take square root: x ≈ ±4.81
Therefore, when y = 8, the points (x, y) ≈ (4.81, 8) and (x, y) ≈ (-4.81, 8) are indeed on the conchoid.