The centre of a circle is (2x - 1, 3x + 1). Find x if the circle passes through (-3, 1) and the length of diameter is 20 unit.

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Identify the center of the circle and the point on the circle
The center of the circle is given as \( (2x - 1, 3x + 1) \) and the point through which the circle passes is \( (-3, 1) \).

### Step 2: Calculate the radius of the circle
The diameter of the circle is given as 20 units. Therefore, the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{20}{2} = 10 \text{ units} \]

### Step 3: Use the distance formula
The distance between the center of the circle and the point on the circle is equal to the radius. We will use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, the distance \( d \) is equal to the radius (10 units), so we set up the equation:
\[ \sqrt{((-3) - (2x - 1))^2 + (1 - (3x + 1))^2} = 10 \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ ((-3) - (2x - 1))^2 + (1 - (3x + 1))^2 = 100 \]

### Step 5: Simplify the expressions inside the squares
First, simplify \( -3 - (2x - 1) \):
\[ -3 - 2x + 1 = -2 - 2x \]
So,
\[ (-2 - 2x)^2 = (2 + 2x)^2 = 4(1 + x)^2 \]

Next, simplify \( 1 - (3x + 1) \):
\[ 1 - 3x - 1 = -3x \]
So,
\[ (-3x)^2 = 9x^2 \]

### Step 6: Substitute back into the equation
Now substitute back into the equation:
\[ (2 + 2x)^2 + (3x)^2 = 100 \]
Expanding gives:
\[ 4(1 + x)^2 + 9x^2 = 100 \]

### Step 7: Expand and simplify
Expanding \( 4(1 + x)^2 \):
\[ 4(1 + 2x + x^2) = 4 + 8x + 4x^2 \]
So the equation becomes:
\[ 4 + 8x + 4x^2 + 9x^2 = 100 \]
Combine like terms:
\[ 13x^2 + 8x + 4 - 100 = 0 \]
This simplifies to:
\[ 13x^2 + 8x - 96 = 0 \]

### Step 8: Solve the quadratic equation
Now, we will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 13, b = 8, c = -96 \):
\[ x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 13 \cdot (-96)}}{2 \cdot 13} \]
Calculating the discriminant:
\[ 8^2 = 64, \quad -4 \cdot 13 \cdot -96 = 4992 \]
So,
\[ b^2 - 4ac = 64 + 4992 = 5056 \]
Now substituting back:
\[ x = \frac{-8 \pm \sqrt{5056}}{26} \]

### Step 9: Calculate the values of x
Calculating \( \sqrt{5056} \) gives approximately \( 71.1 \):
\[ x = \frac{-8 \pm 71.1}{26} \]
This gives two possible solutions:
1. \( x = \frac{63.1}{26} \approx 2.43 \)
2. \( x = \frac{-79.1}{26} \approx -3.05 \)

### Final Answer
The values of \( x \) that satisfy the conditions of the problem are approximately \( x \approx 2.43 \) and \( x \approx -3.05 \).

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