Q. The number of tangents drawn to the curve `xy = 4` from point `(0, 1)` is

Options

A. ` y-(1)/(2)=(1)/(16) (x+8)`
B. `y-(1)/(2)=(1)/(16)(x-8)`
C. ` y+(1)/(2)=-(1)/(16) (x-8)`
D. `y-8=-(1)/(16)(x-(1)/(2))`

Answer

To find the number of tangents drawn to the curve \(xy = 4\) from the point \((0, 1)\), we can follow these steps:

### Step 1: Understand the Curve
The equation \(xy = 4\) represents a rectangular hyperbola. We can rewrite it in terms of \(y\):
\[ y = \frac{4}{x} \]
This shows that for each \(x\), there is a corresponding \(y\) value.

### Step 2: Find the Slope of the Tangent
To find the slope of the tangent to the curve, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = -\frac{4}{x^2} \]
This derivative indicates that the slope of the tangent line is always negative for all \(x \neq 0\).

### Step 3: Equation of the Tangent Line
The general equation of a tangent line at a point \((x_1, y_1)\) on the curve can be expressed using the point-slope form:
\[ y - y_1 = m(x - x_1) \]
where \(m\) is the slope at that point. Since \(y_1 = \frac{4}{x_1}\) and \(m = -\frac{4}{x_1^2}\), the equation becomes:
\[ y - \frac{4}{x_1} = -\frac{4}{x_1^2}(x - x_1) \]

### Step 4: Substitute the Point (0, 1)
Now we substitute the point \((0, 1)\) into the tangent line equation:
\[ 1 - \frac{4}{x_1} = -\frac{4}{x_1^2}(0 - x_1) \]
This simplifies to:
\[ 1 - \frac{4}{x_1} = \frac{4}{x_1} \]
Combining terms gives:
\[ 1 = \frac{8}{x_1} \]
Thus, we can solve for \(x_1\):
\[ x_1 = 8 \]

### Step 5: Verify the Tangent
Now we need to check if there are any other values of \(x_1\) that satisfy the tangent condition. Since the slope is always negative, the tangent line can only touch the hyperbola in one direction. Therefore, there can only be one tangent line from the point \((0, 1)\) to the curve \(xy = 4\).

### Conclusion
Thus, the number of tangents drawn to the curve \(xy = 4\) from the point \((0, 1)\) is:
\[ \boxed{1} \]