The length of sub-tangent to the curve `sqrtx+sqrty=3` at the point (4,1) is

Options

A. 2
B. `1/2`
C. `-3`
D. 4

Answer

To find the length of the sub-tangent to the curve \( \sqrt{x} + \sqrt{y} = 3 \) at the point \( (4, 1) \), we will follow these steps:

### Step 1: Differentiate the curve implicitly
We start with the equation of the curve:
\[ \sqrt{x} + \sqrt{y} = 3 \]
Differentiating both sides with respect to \( x \):
\[ \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \frac{dy}{dx} = 0 \]

### Step 2: Solve for \( \frac{dy}{dx} \)
Rearranging the equation gives:
\[ \frac{1}{2\sqrt{y}} \frac{dy}{dx} = -\frac{1}{2\sqrt{x}} \]
Multiplying both sides by \( 2\sqrt{y} \):
\[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \]

### Step 3: Substitute the point \( (4, 1) \)
Now, we substitute \( x = 4 \) and \( y = 1 \) into the derivative:
\[ \frac{dy}{dx} = -\frac{\sqrt{1}}{\sqrt{4}} = -\frac{1}{2} \]

### Step 4: Calculate the length of the sub-tangent
The length of the sub-tangent \( L \) is given by the formula:
\[ L = \frac{y}{\frac{dy}{dx}} \]
Substituting \( y = 1 \) and \( \frac{dy}{dx} = -\frac{1}{2} \):
\[ L = \frac{1}{-\frac{1}{2}} = -2 \]
Since length cannot be negative, we take the absolute value:
\[ L = 2 \]

### Conclusion
The length of the sub-tangent to the curve at the point \( (4, 1) \) is \( 2 \).

---