The length of subtangent to the curve `x^2 + xy + y^2=7` at the point `(1, -3)` is

Options

A. 3
B. 5
C. 15
D. `3//5`

Answer

To find the length of the subtangent to the curve \( x^2 + xy + y^2 = 7 \) at the point \( (1, -3) \), we will follow these steps:

### Step 1: Differentiate the curve
We start with the equation of the curve:
\[ x^2 + xy + y^2 = 7 \]
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(xy) + \frac{d}{dx}(y^2) = \frac{d}{dx}(7) \]
Using the product rule for \( xy \) and the chain rule for \( y^2 \):
\[ 2x + \left( x \frac{dy}{dx} + y \right) + 2y \frac{dy}{dx} = 0 \]

### Step 2: Substitute the point into the derivative
Now, we substitute the point \( (1, -3) \) into the differentiated equation:
\[ 2(1) + \left( 1 \frac{dy}{dx} + (-3) \right) + 2(-3) \frac{dy}{dx} = 0 \]
This simplifies to:
\[ 2 + \frac{dy}{dx} - 3 - 6 \frac{dy}{dx} = 0 \]
Combining like terms:
\[ 2 - 3 + \frac{dy}{dx} - 6 \frac{dy}{dx} = 0 \implies -1 - 5 \frac{dy}{dx} = 0 \]

### Step 3: Solve for \( \frac{dy}{dx} \)
Rearranging gives:
\[ -5 \frac{dy}{dx} = 1 \implies \frac{dy}{dx} = -\frac{1}{5} \]

### Step 4: Calculate the length of the subtangent
The length of the subtangent \( L \) is given by the formula:
\[ L = y \cdot \left| \frac{1}{\frac{dy}{dx}} \right| \]
Substituting \( y = -3 \) and \( \frac{dy}{dx} = -\frac{1}{5} \):
\[ L = -3 \cdot \left| \frac{1}{-\frac{1}{5}} \right| = -3 \cdot 5 = 15 \]

### Final Answer
Thus, the length of the subtangent at the point \( (1, -3) \) is:
\[ \boxed{15} \]

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