The extremities of the diagonal of a rectangle are `(-4, 4)` and `(6,-1)`. A circle circumscribes the rectangle and cuts inercept of length `AB` on the `y`-axis. The length of `AB` is

Options

A. `11`
B. `12`
C. `13`
D. `14`

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Identify the coordinates of the rectangle's diagonal
The extremities of the diagonal of the rectangle are given as points \( A(-4, 4) \) and \( B(6, -1) \).

### Step 2: Find the center of the circle
The center of the circle that circumscribes the rectangle is the midpoint of the diagonal \( AB \). The midpoint \( M \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \( A \) and \( B \):
\[ M = \left( \frac{-4 + 6}{2}, \frac{4 + (-1)}{2} \right) = \left( \frac{2}{2}, \frac{3}{2} \right) = (1, 1.5) \]

### Step 3: Calculate the radius of the circle
The radius \( r \) of the circle is half the length of the diagonal \( AB \). The length of the diagonal can be found using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ AB = \sqrt{(6 - (-4))^2 + (-1 - 4)^2} = \sqrt{(6 + 4)^2 + (-5)^2} = \sqrt{10^2 + (-5)^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \]
Thus, the radius \( r \) is:
\[ r = \frac{5\sqrt{5}}{2} \]

### Step 4: Write the equation of the circle
The equation of a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 1 \), \( k = 1.5 \), and \( r = \frac{5\sqrt{5}}{2} \):
\[ (x - 1)^2 + \left(y - \frac{3}{2}\right)^2 = \left(\frac{5\sqrt{5}}{2}\right)^2 \]
Calculating \( r^2 \):
\[ r^2 = \frac{25 \cdot 5}{4} = \frac{125}{4} \]
Thus, the equation of the circle becomes:
\[ (x - 1)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{125}{4} \]

### Step 5: Find the intercepts on the y-axis
To find the intercepts on the y-axis, we set \( x = 0 \) in the equation of the circle:
\[ (0 - 1)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{125}{4} \]
This simplifies to:
\[ 1 + \left(y - \frac{3}{2}\right)^2 = \frac{125}{4} \]
Subtracting 1 from both sides:
\[ \left(y - \frac{3}{2}\right)^2 = \frac{125}{4} - 1 = \frac{125}{4} - \frac{4}{4} = \frac{121}{4} \]
Taking the square root:
\[ y - \frac{3}{2} = \pm \frac{11}{2} \]
Thus, we have two values for \( y \):
\[ y = \frac{3}{2} + \frac{11}{2} = 7 \quad \text{and} \quad y = \frac{3}{2} - \frac{11}{2} = -4 \]

### Step 6: Calculate the length of \( AB \)
The length of \( AB \) is the distance between the two y-intercepts:
\[ AB = |y_1 - y_2| = |7 - (-4)| = |7 + 4| = 11 \]

### Final Answer
The length of \( AB \) is \( 11 \).

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