The radius of curvature of the curve `y^2=4x` at the point `(1, 2)` is

Options

A. `4sqrt2`
B. `2sqrt2`
C. `1/sqrt2`
D. `sqrt2`

Answer

To find the radius of curvature of the curve defined by the equation \(y^2 = 4x\) at the point \((1, 2)\), we can follow these steps:

### Step 1: Differentiate the equation to find \(y'\)

Given the equation:

\[ y^2 = 4x \]

We differentiate both sides with respect to \(x\):

\[ 2y \frac{dy}{dx} = 4 \]

This can be rewritten as:

\[ \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y} \]

### Step 2: Evaluate \(y'\) at the point \((1, 2)\)

Now, we substitute \(y = 2\) into the derivative:

\[ y' = \frac{2}{2} = 1 \]

### Step 3: Differentiate again to find \(y''\)

Next, we differentiate \(y'\) to find \(y''\):

\[ y' = \frac{2}{y} \]

Using the quotient rule:

\[ y'' = \frac{(0 \cdot y - 2 \cdot \frac{dy}{dx})}{y^2} \]

Substituting \(y' = \frac{2}{y}\) into the derivative gives:

\[ y'' = \frac{-2y'}{y^2} \]

Substituting \(y' = 1\) and \(y = 2\):

\[ y'' = \frac{-2 \cdot 1}{2^2} = \frac{-2}{4} = -\frac{1}{2} \]

### Step 4: Use the radius of curvature formula

The formula for the radius of curvature \(R\) is given by:

\[ R = \frac{(1 + (y')^2)^{3/2}}{|y''|} \]

Substituting \(y' = 1\) and \(y'' = -\frac{1}{2}\):

\[ R = \frac{(1 + 1^2)^{3/2}}{\left| -\frac{1}{2} \right|} = \frac{(1 + 1)^{3/2}}{\frac{1}{2}} = \frac{(2)^{3/2}}{\frac{1}{2}} = \frac{2\sqrt{2}}{\frac{1}{2}} = 4\sqrt{2} \]

### Final Answer

Thus, the radius of curvature at the point \((1, 2)\) is:

\[ R = 4\sqrt{2} \]

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