The maximum distance of the point (a, 0) from the curve `2x^(2) + y^(2) - 2x = 0` is -

Options

A. `sqrt((1 - 2 a + a^(2)))`
B. `sqrt((1 + 2a + 2a^(2)))`
C. ` sqrt((1 + 2a - a^(2)))`
D. `sqrt((1 - 2a + 2a^(2)))`

Answer

To find the maximum distance of the point (a, 0) from the curve given by the equation \(2x^2 + y^2 - 2x = 0\), we will follow these steps:

### Step 1: Rearranging the Curve Equation
The given equation of the curve is:
\[ 2x^2 + y^2 - 2x = 0 \]
We can rearrange this to express \(y^2\) in terms of \(x\):
\[ y^2 = 2x - 2x^2 \]

### Step 2: Expressing the Points on the Curve
From the equation \(y^2 = 2x(1 - x)\), we can express \(y\) as:
\[ y = \pm \sqrt{2x(1 - x)} \]
Thus, any point \(P\) on the curve can be represented as:
\[ P(x, \sqrt{2x(1 - x)}) \quad \text{or} \quad P(x, -\sqrt{2x(1 - x)}) \]

### Step 3: Finding the Distance from the Point (a, 0)
The distance \(D\) from the point \((a, 0)\) to the point \(P(x, y)\) on the curve is given by the distance formula:
\[ D = \sqrt{(x - a)^2 + y^2} \]
Substituting \(y^2\) from our earlier expression:
\[ D = \sqrt{(x - a)^2 + 2x(1 - x)} \]

### Step 4: Simplifying the Distance Formula
Now, we can simplify the expression for \(D^2\):
\[ D^2 = (x - a)^2 + 2x(1 - x) \]
Expanding this:
\[ D^2 = (x^2 - 2ax + a^2) + (2x - 2x^2) \]
Combining like terms:
\[ D^2 = -x^2 - 2ax + 2x + a^2 \]

### Step 5: Finding the Maximum Distance
To find the maximum distance, we need to maximize \(D^2\). We can differentiate \(D^2\) with respect to \(x\) and set the derivative to zero:
\[ \frac{d(D^2)}{dx} = -2x - 2a + 2 = 0 \]
Solving for \(x\):
\[ -2x - 2a + 2 = 0 \implies 2x = 2 - 2a \implies x = 1 - a \]

### Step 6: Substituting Back to Find Maximum Distance
Now, we substitute \(x = 1 - a\) back into the expression for \(D^2\):
\[ D^2 = -(1 - a)^2 - 2a(1 - a) + a^2 \]
Calculating:
\[ D^2 = -(1 - 2a + a^2) - 2a + 2a^2 + a^2 \]
\[ D^2 = -1 + 2a - a^2 - 2a + 2a^2 + a^2 = -1 + 2a^2 \]

### Step 7: Finding the Maximum Distance
The maximum distance \(D\) is given by:
\[ D = \sqrt{-1 + 2a^2} \]

### Final Answer
Thus, the maximum distance of the point \((a, 0)\) from the curve is:
\[ D = \sqrt{2a^2 - 1} \]