The tangent at any point on the curve xy = 4 makes intercepls on the coordinates axes as a and b. Then the value of ab is

Options

A. 8
B. 16
C. 32
D. 64

Answer

To solve the problem, we need to find the value of \( ab \) where \( a \) and \( b \) are the x-intercept and y-intercept of the tangent line to the curve \( xy = 4 \) at any point \( (h, k) \) on the curve.

### Step-by-step Solution:

1. **Identify the point on the curve:**
The curve is given by the equation \( xy = 4 \). Let’s take a point \( (h, k) \) on this curve. Therefore, we have:
\[ hk = 4 \quad \text{(1)} \]

2. **Differentiate the curve:**
To find the slope of the tangent line at the point \( (h, k) \), we differentiate the equation \( xy = 4 \) implicitly with respect to \( x \):
\[ x \frac{dy}{dx} + y = 0 \]
Rearranging gives:
\[ \frac{dy}{dx} = -\frac{y}{x} \]
Substituting \( y = k \) and \( x = h \) gives the slope of the tangent at point \( (h, k) \):
\[ \frac{dy}{dx} = -\frac{k}{h} \]

3. **Equation of the tangent line:**
The equation of the tangent line at point \( (h, k) \) can be written using the point-slope form:
\[ y - k = \left(-\frac{k}{h}\right)(x - h) \]
Simplifying this gives:
\[ y - k = -\frac{k}{h}x + k \]
Rearranging leads to:
\[ kx + hy = 2hk \]

4. **Finding the intercepts:**
The intercepts can be found from the equation \( kx + hy = 2hk \):
- **x-intercept (where \( y = 0 \)):**
\[ kx = 2hk \implies x = \frac{2h}{1} = \frac{2h}{k} \quad \text{(let this be } a \text{)} \]
- **y-intercept (where \( x = 0 \)):**
\[ hy = 2hk \implies y = \frac{2hk}{h} = 2k \quad \text{(let this be } b \text{)} \]

5. **Expressing \( ab \):**
Now, we have:
\[ a = \frac{8}{k} \quad \text{and} \quad b = \frac{8}{h} \]
Therefore, the product \( ab \) is:
\[ ab = \left(\frac{8}{k}\right) \left(\frac{8}{h}\right) = \frac{64}{kh} \]

6. **Substituting \( kh \):**
From equation (1), we know \( kh = 4 \). Thus:
\[ ab = \frac{64}{4} = 16 \]

### Final Answer:
The value of \( ab \) is \( \boxed{16} \).