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 on the curve.

### Step-by-step Solution:

1. **Identify the Point on the Curve**:
Let \( P(h, k) \) be a point on the curve \( xy = 4 \). Since \( P \) lies on the curve, it satisfies the equation:
\[ hk = 4 \quad \text{(Equation 1)} \]

2. **Differentiate the Curve**:
Differentiate the equation \( xy = 4 \) with respect to \( x \):
\[ \frac{d}{dx}(xy) = \frac{d}{dx}(4) \]
Using the product rule, we get:
\[ y + x\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{y}{x} \]
Substituting \( x = h \) and \( y = k \), the slope of the tangent at point \( P \) is:
\[ \frac{dy}{dx} = -\frac{k}{h} \]

3. **Equation of the Tangent Line**:
The equation of the tangent line at point \( P(h, k) \) can be written using the point-slope form:
\[ y - k = -\frac{k}{h}(x - h) \]
Rearranging this gives:
\[ y - k = -\frac{k}{h}x + k \implies y = -\frac{k}{h}x + 2k \]

4. **Finding the Intercepts**:
To find the x-intercept (\( a \)), set \( y = 0 \):
\[ 0 = -\frac{k}{h}x + 2k \implies \frac{k}{h}x = 2k \implies x = \frac{2kh}{k} = 2h \implies a = 2h \]
To find the y-intercept (\( b \)), set \( x = 0 \):
\[ y = -\frac{k}{h}(0) + 2k = 2k \implies b = 2k \]

5. **Calculating \( ab \)**:
Now, we calculate \( ab \):
\[ ab = (2h)(2k) = 4hk \]
From Equation 1, we know \( hk = 4 \):
\[ ab = 4 \times 4 = 16 \]

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