The length of the Sub tangent at `(2,2)` to the curve `x^5 = 2y^4` is

Options

A. `5//2`
B. `8//5`
C. `2//5`
D. `5//8`

Answer

To find the length of the subtangent at the point (2, 2) to the curve given by the equation \( x^5 = 2y^4 \), we will follow these steps:

### Step 1: Understand the formula for the length of the subtangent
The length of the subtangent at a point \((x_1, y_1)\) on the curve is given by the formula:
\[ \text{Length of subtangent} = |y_1 \cdot \frac{dx}{dy}| \]

### Step 2: Differentiate the curve equation
We start with the curve equation:
\[ x^5 = 2y^4 \]
To find \(\frac{dx}{dy}\), we differentiate both sides with respect to \(y\).

Differentiating the left side:
\[ \frac{d}{dy}(x^5) = 5x^4 \cdot \frac{dx}{dy} \]

Differentiating the right side:
\[ \frac{d}{dy}(2y^4) = 8y^3 \]

Setting these equal gives us:
\[ 5x^4 \cdot \frac{dx}{dy} = 8y^3 \]

### Step 3: Solve for \(\frac{dx}{dy}\)
Rearranging the equation to solve for \(\frac{dx}{dy}\):
\[ \frac{dx}{dy} = \frac{8y^3}{5x^4} \]

### Step 4: Substitute the point (2, 2)
Now we substitute \(x = 2\) and \(y = 2\) into the expression for \(\frac{dx}{dy}\):
\[ \frac{dx}{dy} = \frac{8(2)^3}{5(2)^4} = \frac{8 \cdot 8}{5 \cdot 16} = \frac{64}{80} = \frac{8}{10} = \frac{4}{5} \]

### Step 5: Calculate the length of the subtangent
Now we can substitute \(y_1 = 2\) and \(\frac{dx}{dy} = \frac{4}{5}\) into the formula for the length of the subtangent:
\[ \text{Length of subtangent} = |2 \cdot \frac{4}{5}| = \frac{8}{5} \]

### Final Answer
Thus, the length of the subtangent at the point (2, 2) is:
\[ \boxed{\frac{8}{5}} \]

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