The equation of the line of the shortest distance between the parabola `y^(2)=4x` and the circle `x^(2)+y^(2)-4x-2y+4=0` is

Options

A. x + y = 3
B. x - y = 3
C. 2x + y = 5
D. None of these

Answer

To find the equation of the line of the shortest distance between the parabola \( y^2 = 4x \) and the circle given by \( x^2 + y^2 - 4x - 2y + 4 = 0 \), we can follow these steps:

### Step 1: Identify the center and radius of the circle
The equation of the circle can be rewritten in standard form. We start with:
\[ x^2 + y^2 - 4x - 2y + 4 = 0 \]
We can complete the square for both \( x \) and \( y \).

For \( x \):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

For \( y \):
\[ y^2 - 2y = (y - 1)^2 - 1 \]

Substituting these back into the equation gives:
\[ (x - 2)^2 - 4 + (y - 1)^2 - 1 + 4 = 0 \]
\[ (x - 2)^2 + (y - 1)^2 - 1 = 0 \]
\[ (x - 2)^2 + (y - 1)^2 = 1 \]

From this, we can see that the center of the circle is \( (2, 1) \) and the radius is \( r = 1 \).

### Step 2: Identify the normal to the parabola
The parabola \( y^2 = 4x \) can be expressed in parametric form as:
\[ P(t) = (t^2, 2t) \]
The slope of the tangent at this point is given by:
\[ \frac{dy}{dx} = \frac{2}{2t} = \frac{1}{t} \]
Thus, the slope of the normal line is:
\[ -\frac{1}{\frac{1}{t}} = -t \]

The equation of the normal line at point \( P(t) \) is:
\[ y - 2t = -t(x - t^2) \]
Rearranging gives:
\[ y = -tx + t^2 + 2t \]

### Step 3: Find the intersection with the circle
The normal line must pass through the center of the circle \( (2, 1) \). Therefore, substituting \( x = 2 \) and \( y = 1 \) into the normal line equation:
\[ 1 = -t(2) + t^2 + 2t \]
Simplifying gives:
\[ 1 = t^2 + 0t \]
Thus:
\[ t^2 - 1 = 0 \implies (t - 1)(t + 1) = 0 \]
This gives us \( t = 1 \) or \( t = -1 \).

### Step 4: Determine the line equations
Using \( t = 1 \):
\[ y - 2 = -1(x - 1^2) \implies y - 2 = -x + 1 \implies y + x - 3 = 0 \]

Using \( t = -1 \):
\[ y - (-2) = 1(x - 1) \implies y + 2 = x - 1 \implies x - y - 3 = 0 \]

### Step 5: Conclusion
The equation of the line of the shortest distance between the parabola and the circle is:
\[ x + y - 3 = 0 \]