To find the other extremity of the diameter of the circle given that one extremity is (2, 3), we can follow these steps:
### Step 1: Identify the center of the circle
The equation of the circle is given as:
\[ x^2 + y^2 - 5x - 8y + 21 = 0 \]
To find the center, we need to rewrite the equation in the standard form. We can do this by completing the square for both \(x\) and \(y\).
1. Rearranging the equation:
\[ x^2 - 5x + y^2 - 8y + 21 = 0 \]
2. Completing the square for \(x\):
\[ x^2 - 5x = (x - \frac{5}{2})^2 - \frac{25}{4} \]
3. Completing the square for \(y\):
\[ y^2 - 8y = (y - 4)^2 - 16 \]
4. Substituting back into the equation:
\[ (x - \frac{5}{2})^2 - \frac{25}{4} + (y - 4)^2 - 16 + 21 = 0 \]
\[ (x - \frac{5}{2})^2 + (y - 4)^2 - \frac{25}{4} - 16 + 21 = 0 \]
\[ (x - \frac{5}{2})^2 + (y - 4)^2 = \frac{25}{4} + 16 - 21 \]
\[ (x - \frac{5}{2})^2 + (y - 4)^2 = \frac{25}{4} + \frac{64}{4} - \frac{84}{4} \]
\[ (x - \frac{5}{2})^2 + (y - 4)^2 = \frac{5}{4} \]
From this, we can see that the center of the circle is at:
\[ \left(\frac{5}{2}, 4\right) \]
### Step 2: Use the midpoint formula
Since (2, 3) is one extremity of the diameter, we can denote the other extremity as (a, b). The midpoint of the diameter can be calculated using the midpoint formula:
\[ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Where \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (a, b) \).
Setting the midpoint equal to the center of the circle:
\[ \left(\frac{2 + a}{2}, \frac{3 + b}{2}\right) = \left(\frac{5}{2}, 4\right) \]
### Step 3: Set up equations
From the midpoint equality, we can set up two equations:
1. For the x-coordinates:
\[ \frac{2 + a}{2} = \frac{5}{2} \]
Multiplying both sides by 2:
\[ 2 + a = 5 \]
Thus,
\[ a = 5 - 2 = 3 \]
2. For the y-coordinates:
\[ \frac{3 + b}{2} = 4 \]
Multiplying both sides by 2:
\[ 3 + b = 8 \]
Thus,
\[ b = 8 - 3 = 5 \]
### Step 4: Conclusion
The coordinates of the other extremity of the diameter are:
\[ (a, b) = (3, 5) \]
Therefore, the other extremity of the diameter is \( (3, 5) \).
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