Find the point on the parabola `y=(x-3)^2,` where the tangent is parabola to the line joining (3,0) and (4,1

Explanation

To find the point on the parabola \( y = (x - 3)^2 \) where the tangent is parallel to the line joining the points \( (3, 0) \) and \( (4, 1) \), we can follow these steps:

### Step 1: Find the slope of the line joining the points (3, 0) and (4, 1)

The slope \( m \) of a line joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the given points \( (3, 0) \) and \( (4, 1) \):

\[ m = \frac{1 - 0}{4 - 3} = \frac{1}{1} = 1 \]

### Step 2: Find the derivative of the parabola

The equation of the parabola is \( y = (x - 3)^2 \). To find the slope of the tangent to the parabola at any point, we need to differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = 2(x - 3) \]

### Step 3: Set the derivative equal to the slope of the line

Since we want the tangent to be parallel to the line, we set the derivative equal to the slope we found in Step 1:

\[ 2(x - 3) = 1 \]

### Step 4: Solve for \( x \)

Now, we solve for \( x \):

\[ 2(x - 3) = 1 \\ x - 3 = \frac{1}{2} \\ x = \frac{1}{2} + 3 = \frac{7}{2} \]

### Step 5: Find the corresponding \( y \) value on the parabola

Now that we have \( x = \frac{7}{2} \), we can find the corresponding \( y \) value by substituting \( x \) back into the equation of the parabola:

\[ y = \left(\frac{7}{2} - 3\right)^2 \\ y = \left(\frac{7}{2} - \frac{6}{2}\right)^2 \\ y = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Conclusion

Thus, the point on the parabola where the tangent is parallel to the line joining the points \( (3, 0) \) and \( (4, 1) \) is:

\[ \left(\frac{7}{2}, \frac{1}{4}\right) \]