The length of the subtangent to the curve `x^2+xy+y^2=7` at `(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: Find the derivative \(\frac{dy}{dx}\)
Given the curve:
\[ x^2 + xy + y^2 = 7 \]
we will differentiate both sides implicitly with respect to \(x\).

Differentiating:
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(xy) + \frac{d}{dx}(y^2) = \frac{d}{dx}(7) \]
This gives us:
\[ 2x + \left(x \frac{dy}{dx} + y\right) + 2y \frac{dy}{dx} = 0 \]
Rearranging, we have:
\[ 2x + y + (x + 2y) \frac{dy}{dx} = 0 \]
Now, isolate \(\frac{dy}{dx}\):
\[ (x + 2y) \frac{dy}{dx} = -2x - y \]
Thus,
\[ \frac{dy}{dx} = \frac{-2x - y}{x + 2y} \]

### Step 2: Substitute the point \((1, -3)\) into the derivative
Now, we will substitute \(x = 1\) and \(y = -3\):
\[ \frac{dy}{dx} = \frac{-2(1) - (-3)}{1 + 2(-3)} = \frac{-2 + 3}{1 - 6} = \frac{1}{-5} = -\frac{1}{5} \]

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

### Final Answer
The length of the subtangent to the curve at the point \((1, -3)\) is \(15\).

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