A (1 ,0 ) and B (7 ,0 ) are two points on the axis o f x. A point P is taken in the first quadrant such that PAB is an isosceles triangle and PB = 5 units. Find the equation of the circle described on PA as diameter.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the Points
Let the points A and B be defined as:
- A(1, 0)
- B(7, 0)

Let the point P in the first quadrant be defined as P(x, y).

### Step 2: Set Up the Isosceles Triangle Condition
Since triangle PAB is isosceles, we have:
\[ PA = PB \]

Using the distance formula, we can express these distances:
- Distance PA:
\[ PA = \sqrt{(x - 1)^2 + (y - 0)^2} = \sqrt{(x - 1)^2 + y^2} \]

- Distance PB:
\[ PB = \sqrt{(x - 7)^2 + (y - 0)^2} = \sqrt{(x - 7)^2 + y^2} \]

Setting these equal gives us:
\[ \sqrt{(x - 1)^2 + y^2} = \sqrt{(x - 7)^2 + y^2} \]

### Step 3: Square Both Sides
Squaring both sides to eliminate the square roots:
\[ (x - 1)^2 + y^2 = (x - 7)^2 + y^2 \]

### Step 4: Simplify the Equation
Cancel \( y^2 \) from both sides:
\[ (x - 1)^2 = (x - 7)^2 \]

Expanding both sides:
\[ x^2 - 2x + 1 = x^2 - 14x + 49 \]

### Step 5: Rearranging the Equation
Subtract \( x^2 \) from both sides:
\[ -2x + 1 = -14x + 49 \]

Rearranging gives:
\[ 12x = 48 \implies x = 4 \]

### Step 6: Use the Distance PB = 5 Condition
Now we know \( x = 4 \). We can use the condition \( PB = 5 \):
\[ PB = \sqrt{(4 - 7)^2 + y^2} = 5 \]

### Step 7: Square Both Sides Again
Squaring both sides:
\[ (4 - 7)^2 + y^2 = 25 \]
\[ 9 + y^2 = 25 \]

### Step 8: Solve for y
Subtract 9 from both sides:
\[ y^2 = 16 \implies y = 4 \quad (\text{since P is in the first quadrant}) \]

Thus, the coordinates of point P are:
\[ P(4, 4) \]

### Step 9: Find the Center of the Circle
The circle is described on PA as the diameter. The midpoint O of PA is given by:
\[ O\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = O\left(\frac{1 + 4}{2}, \frac{0 + 4}{2}\right) = O\left(\frac{5}{2}, 2\right) \]

### Step 10: Find the Radius
The radius \( r \) is half the length of PA:
\[ PA = \sqrt{(4 - 1)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, the radius \( r = \frac{5}{2} \).

### Step 11: Write the Equation of the Circle
The equation of the circle with center O and radius r is:
\[ \left(x - \frac{5}{2}\right)^2 + (y - 2)^2 = \left(\frac{5}{2}\right)^2 \]
Expanding this:
\[ \left(x - \frac{5}{2}\right)^2 + (y - 2)^2 = \frac{25}{4} \]

### Step 12: Rearranging the Circle Equation
Expanding and rearranging gives:
\[ x^2 - 5x + \frac{25}{4} + y^2 - 4y + 4 = \frac{25}{4} \]
This simplifies to:
\[ x^2 + y^2 - 5x - 4y + 4 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 5x - 4y + 4 = 0 \]

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