What is the y co-ordinate of the centre of curvature for the curve whose parametric
representation is x = ct , y = c/t

To find the \( y \)-coordinate of the center of curvature for the curve given by the parametric equations:

\[
x = ct \quad \text{and} \quad y = \frac{c}{t}
\]

we need to use the formula for the coordinates of the center of curvature \((x_c, y_c)\) in parametric form. The center of curvature is located at a point where the normal to the curve at a given point intersects the osculating circle. The formula for the center of curvature is given by:

\[
y_c = y + \frac{\left( \dot{x}^2 + \dot{y}^2 \right)^{3/2}}{\dot{x} \ddot{y} - \dot{y} \ddot{x}}
\]

Where:
- \(\dot{x} = \frac{dx}{dt}\)
- \(\dot{y} = \frac{dy}{dt}\)
- \(\ddot{x} = \frac{d^2x}{dt^2}\)
- \(\ddot{y} = \frac{d^2y}{dt^2}\)

### Step 1: Compute the first and second derivatives of \(x(t)\) and \(y(t)\)

Given \(x = ct\) and \(y = \frac{c}{t}\), let's find the derivatives:

\[
\dot{x} = \frac{dx}{dt} = c
\]
\[
\dot{y} = \frac{dy}{dt} = -\frac{c}{t^2}
\]

Now the second derivatives:

\[
\ddot{x} = \frac{d^2x}{dt^2} = 0
\]
\[
\ddot{y} = \frac{d^2y}{dt^2} = \frac{2c}{t^3}
\]

### Step 2: Apply the formula for \(y_c\)

Now substitute these values into the formula for \(y_c\):

\[
y_c = \frac{c}{t} + \frac{\left( c^2 + \left(-\frac{c}{t^2}\right)^2 \right)^{3/2}}{c \cdot \frac{2c}{t^3} - \left( -\frac{c}{t^2} \right) \cdot 0}
\]

Simplifying:

\[
y_c = \frac{c}{t} + \frac{\left( c^2 + \frac{c^2}{t^4} \right)^{3/2}}{2c^2/t^3}
\]

Simplify further:

\[
y_c = \frac{c}{t} + \frac{c^3 \left( 1 + \frac{1}{t^4} \right)^{3/2}}{2c^2/t^3}
\]

\[
y_c = \frac{c}{t} + \frac{t^3 \cdot c \left( 1 + \frac{1}{t^4} \right)^{3/2}}{2}
\]

This is the expression for the \(y\)-coordinate of the center of curvature.

Would you like more details or have any questions?

### 5 Related Questions:
1. How do we interpret the center of curvature geometrically?
2. What is the significance of curvature in parametric equations?
3. Can this method be applied to non-parametric equations as well?
4. How does the center of curvature relate to the radius of curvature?
5. How do we find the radius of curvature for a given parametric curve?

### Tip:
For curves given in parametric form, always compute derivatives carefully as they are key to finding curvature and related properties like the center of curvature.

What is the y-coordinate of the centre of curvature for the curve whose parametric representation is x = ct , y = c/t?

Learn how to compute the y-coordinate of the centre of curvature for the parametric curve given by x = ct and y = c/t. This problem involves parametric equations, curvature, and differential calculus, suitable for students in Grades 11-12.

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