If the line `x-ay=5` is a chord of the parabola `y^(2) = 20x,` then the circle with this chord as diamerte will always touch the line

Options

A. `x+5=0`
B. `2y+5=0`
C. `2x+5=y`
D. depend on a

Answer

To solve the problem step by step, we will analyze the given information and apply the properties of the parabola and the circle.

### Step 1: Understand the Parabola
The equation of the parabola given is \( y^2 = 20x \). This can be compared to the standard form of a parabola \( y^2 = 4ax \), where \( a \) is the distance from the vertex to the focus.

**Calculation:**
From \( y^2 = 20x \), we can identify that \( 4a = 20 \), thus:
\[ a = \frac{20}{4} = 5 \]

### Step 2: Find the Directrix
The directrix of a parabola given by \( y^2 = 4ax \) is given by the equation \( x = -a \).

**Calculation:**
Since we found \( a = 5 \):
\[ \text{Directrix: } x = -5 \]

### Step 3: Analyze the Chord
The line given is \( x - ay = 5 \). We can rearrange this to express \( y \):
\[ ay = x - 5 \implies y = \frac{x - 5}{a} \]

### Step 4: Determine the Circle
The circle with the chord as a diameter will have its center at the midpoint of the endpoints of the chord. The endpoints of the chord can be found by substituting the \( y \) values from the chord equation into the parabola equation.

### Step 5: Find the Points of Intersection
Substituting \( y = \frac{x - 5}{a} \) into the parabola equation \( y^2 = 20x \):
\[ \left(\frac{x - 5}{a}\right)^2 = 20x \]
Multiplying through by \( a^2 \):
\[ (x - 5)^2 = 20ax \]
Expanding and rearranging:
\[ x^2 - 10x + 25 = 20ax \implies x^2 - (10 + 20a)x + 25 = 0 \]

### Step 6: Use the Quadratic Formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -(10 + 20a), c = 25 \):
\[ x = \frac{10 + 20a \pm \sqrt{(10 + 20a)^2 - 100}}{2} \]

### Step 7: Find the Circle's Center
The center of the circle can be calculated as the average of the x-coordinates of the intersection points. The radius can be calculated as half the distance between these points.

### Step 8: Show that the Circle Touches the Line
To show that the circle touches the line \( x + 5 = 0 \) (the directrix), we need to prove that the distance from the center of the circle to the line equals the radius of the circle.

### Conclusion
Since the circle is constructed with the chord of the parabola as its diameter, and the properties of the parabola dictate that the directrix is a tangent to the circle, we conclude that the circle will always touch the line \( x + 5 = 0 \).