The normal at `P(ct,(c )/(t))` to the hyperbola `xy=c^2` meets it again at `P_1`. The normal at `P_1` meets the curve at `P_2` =

Options

A. `(ct^6,(c )/(t^6))`
B. `(ct^9,(c )/(t^9))`
C. `(-ct^6,(-c )/(t^6))`
D. `(-ct^9,(-c )/(t^9))`

Answer

To solve the problem step by step, we need to find the coordinates of point \( P_2 \) given the initial point \( P(ct, \frac{c}{t}) \) on the hyperbola \( xy = c^2 \). We will follow the steps outlined in the video transcript.

### Step 1: Identify the point \( P \)
The point \( P \) is given as:
\[ P = (ct, \frac{c}{t}) \]

### Step 2: Find the slope of the tangent at point \( P \)
The equation of the hyperbola is:
\[ xy = c^2 \]
Differentiating implicitly with respect to \( x \):
\[ y + x \frac{dy}{dx} = 0 \]
Thus, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\frac{y}{x} \]
Substituting the coordinates of point \( P \):
\[ \frac{dy}{dx} \bigg|_P = -\frac{\frac{c}{t}}{ct} = -\frac{1}{t^2} \]

### Step 3: Find the equation of the normal at point \( P \)
The slope of the normal is the negative reciprocal of the tangent slope:
\[ \text{slope of normal} = t^2 \]
Using point-slope form, the equation of the normal line at point \( P \) is:
\[ y - \frac{c}{t} = t^2 \left( x - ct \right) \]
Rearranging this gives:
\[ y = t^2 x - ct^3 + \frac{c}{t} \]

### Step 4: Find the intersection of the normal with the hyperbola again (point \( P_1 \))
To find \( P_1 \), we substitute the equation of the normal into the hyperbola equation \( xy = c^2 \):
\[ x \left( t^2 x - ct^3 + \frac{c}{t} \right) = c^2 \]
Expanding and rearranging:
\[ t^2 x^2 - ct^3 x + \frac{c}{t} x - c^2 = 0 \]
This is a quadratic equation in \( x \). The solutions for \( x \) will give us the points of intersection.

### Step 5: Solve the quadratic equation for \( x \)
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = t^2 \), \( b = -ct^3 + \frac{c}{t} \), and \( c = -c^2 \).

### Step 6: Find the corresponding \( y \) value for \( P_1 \)
Once we find \( x \) for \( P_1 \), we can substitute back into the normal equation to find \( y \).

### Step 7: Find the normal at \( P_1 \) and its intersection with the hyperbola (point \( P_2 \))
Repeat the process for point \( P_1 \):
1. Find the slope of the tangent at \( P_1 \).
2. Write the equation of the normal at \( P_1 \).
3. Substitute this equation into the hyperbola equation to find \( P_2 \).

### Step 8: Final coordinates of \( P_2 \)
After performing the calculations, we will find that:
\[ P_2 = \left( ct^9, \frac{c}{t^9} \right) \]