If the tangent at (16,64) on the curve `y^(2) = x^(3)` meets the curve again at Q(u, v) then uv is equal to_____

Answer

To solve the problem, we need to find the value of \( uv \) where \( Q(u, v) \) is the point where the tangent at the point \( (16, 64) \) on the curve \( y^2 = x^3 \) meets the curve again.

### Step-by-Step Solution:

1. **Differentiate the Curve:**
The given curve is \( y^2 = x^3 \). We differentiate it implicitly with respect to \( x \):
\[ 2y \frac{dy}{dx} = 3x^2 \]
Therefore, the slope of the tangent line at any point on the curve is:
\[ \frac{dy}{dx} = \frac{3x^2}{2y} \]

2. **Find the Slope at the Point (16, 64):**
Substitute \( x = 16 \) and \( y = 64 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{(16, 64)} = \frac{3(16^2)}{2(64)} = \frac{3 \cdot 256}{128} = \frac{768}{128} = 6 \]
Thus, the slope of the tangent at the point \( (16, 64) \) is \( 6 \).

3. **Equation of the Tangent Line:**
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) = (16, 64) \) and \( m = 6 \):
\[ y - 64 = 6(x - 16) \]
Simplifying this gives:
\[ y - 64 = 6x - 96 \implies y = 6x - 32 \]

4. **Substituting into the Curve Equation:**
We need to find where this tangent line intersects the curve \( y^2 = x^3 \) again. Substitute \( y = 6x - 32 \) into \( y^2 = x^3 \):
\[ (6x - 32)^2 = x^3 \]
Expanding the left side:
\[ 36x^2 - 384x + 1024 = x^3 \]
Rearranging gives:
\[ x^3 - 36x^2 + 384x - 1024 = 0 \]

5. **Finding the Roots:**
We know one root is \( x = 16 \) (the point of tangency). Let the roots be \( \alpha, \beta, \gamma \) where \( \alpha = \beta = 16 \) and \( \gamma \) is the unknown root. By Vieta's formulas, the sum of the roots is:
\[ \alpha + \beta + \gamma = 36 \implies 16 + 16 + \gamma = 36 \implies \gamma = 4 \]

6. **Finding the Corresponding y-coordinate:**
Now, substitute \( x = 4 \) back into the curve equation to find \( y \):
\[ y^2 = 4^3 = 64 \implies y = 8 \text{ or } y = -8 \]
Thus, the point \( Q \) can be \( (4, 8) \) or \( (4, -8) \).

7. **Calculating \( uv \):**
Since \( u = 4 \) and \( v = 8 \) (or \( v = -8 \)):
\[ uv = 4 \times 8 = 32 \quad \text{(or } uv = 4 \times (-8) = -32\text{)} \]

### Final Answer:
Thus, the value of \( uv \) is \( 32 \).