Find a point on the parabola `y=(x-2)^(2)`, where the tangent is parallel to the chord joining (2, 0) and (4, 4).

Answer

To find a point on the parabola \( y = (x - 2)^2 \) where the tangent is parallel to the chord joining the points \( (2, 0) \) and \( (4, 4) \), we will follow these steps:

### Step 1: Find the slope of the chord
The slope of the chord joining the points \( (2, 0) \) and \( (4, 4) \) can be calculated using the formula for the slope between two points:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the given points:
\[ \text{slope} = \frac{4 - 0}{4 - 2} = \frac{4}{2} = 2 \]

### Step 2: Find the derivative of the parabola
The derivative of the function \( y = (x - 2)^2 \) will give us the slope of the tangent at any point \( x \):
\[ \frac{dy}{dx} = 2(x - 2) \]

### Step 3: Set the derivative equal to the slope of the chord
Since we want the tangent to be parallel to the chord, we set the derivative equal to the slope of the chord:
\[ 2(x - 2) = 2 \]

### Step 4: Solve for \( x \)
Now, we solve the equation:
\[ 2(x - 2) = 2 \]
Dividing both sides by 2:
\[ x - 2 = 1 \]
Adding 2 to both sides:
\[ x = 3 \]

### Step 5: Find the corresponding \( y \) value
Now, we substitute \( x = 3 \) back into the parabola equation to find \( y \):
\[ y = (3 - 2)^2 = 1^2 = 1 \]

### Step 6: State the point
Thus, the point on the parabola where the tangent is parallel to the chord joining the points \( (2, 0) \) and \( (4, 4) \) is:
\[ (3, 1) \]

### Summary
The point on the parabola \( y = (x - 2)^2 \) where the tangent is parallel to the chord joining \( (2, 0) \) and \( (4, 4) \) is \( (3, 1) \).

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