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To solve this problem, let's go through each part systematically.

### Given Information
The points \( A(-3, -2) \), \( B(-6, 0) \), and \( C(1, q) \) lie on the circumference of a circle, with \( \angle BAC = 90^\circ \). Since \( \angle BAC = 90^\circ \), it implies that \( B \), \( A \), and \( C \) form a right angle at \( A \). Therefore, point \( A \) is at the right angle, making \( BC \) the diameter of the circle (since the angle subtended by a diameter on a circle is always a right angle).

Let's tackle each part:

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### Part (a): Find the value of \( q \).

Since \( B \) and \( C \) are endpoints of the diameter of the circle, the midpoint of \( BC \) will be the center of the circle. We can find \( q \) by using the distance formula to ensure that \( B \) and \( C \) are symmetric around the center point.

1. **Find the midpoint of \( B \) and \( C \)**:
   The midpoint \( M \) of \( B(-6, 0) \) and \( C(1, q) \) is:
   \[
   M = \left( \frac{-6 + 1}{2}, \frac{0 + q}{2} \right) = \left( -\frac{5}{2}, \frac{q}{2} \right)
   \]

2. **Ensure that this midpoint \( M \) is equidistant from both \( A \) and the midpoint**:
   Since \( A(-3, -2) \) is the right angle, the perpendicular distances from \( A \) to \( B \) and \( A \) to \( C \) should be equal. Therefore, we can equate

The points A(-3, -2), B(-6, 0), and C(1, q) lie on the circumference of a circle such that angle BAC = 90°. Find the value of q and the equation of the circle.

Solve a problem involving points A(-3, -2), B(-6, 0), and C(1, q) on a circle's circumference with angle BAC = 90°. Learn to find the value of q and the equation of the circle using coordinate geometry concepts, including the midpoint and distance formulas, and Thales' Theorem.

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