Summary of "שאלת מתכונת 4 יחידות-שאלון 471-גיאומטריה עם מעגל."

Problem

Triangle ABC is inscribed in a circle whose center M is at (4, 8). The circle passes through the origin O(0, 0). Point A is the intersection of the circle with the y-axis (so x_A = 0). Side AB is parallel to the x-axis. The slope of BC is 1. Find the circle equation and the coordinates of A, B, and C; give the equation of line BC.

Key results

Circle equation: (x − 4)^2 + (y − 8)^2 = 80.
A = (0, 16) (the other y-axis intersection is O = (0, 0)).
B = (8, 16).
Line BC: y = x + 8.
C = (−4, 4).

Step-by-step solution

Write the circle equation from the center:
- For center (a, b), (x − a)^2 + (y − b)^2 = r^2.
- With center M(4, 8): (x − 4)^2 + (y − 8)^2 = r^2.
Find r^2 using a known point on the circle (the origin O(0, 0)):
- r^2 = (0 − 4)^2 + (0 − 8)^2 = 16 + 64 = 80.
- Thus the circle is (x − 4)^2 + (y − 8)^2 = 80.
Find intersections with the y-axis (x = 0):
- Substitute x = 0: 16 + (y − 8)^2 = 80 → (y − 8)^2 = 64.
- y − 8 = ±8 → y = 16 or y = 0.
- So intersections are (0, 16) and (0, 0). Since (0, 0) is O, the other is A = (0, 16).
Use AB ∥ x-axis to get B:
- AB horizontal ⇒ y_B = y_A = 16.
- Substitute y = 16 into the circle: (x − 4)^2 + 64 = 80 → (x − 4)^2 = 16 → x = 0 or 8.
- x = 0 gives A; x = 8 gives B = (8, 16).
Get line BC from slope and point B:
- Slope m = 1 and point B(8, 16): y − 16 = 1(x − 8) → y = x + 8.
Find C as the other intersection of y = x + 8 with the circle:
- Substitute y = x + 8 into the circle: (x − 4)^2 + x^2 = 80.
- Expand: 2x^2 − 8x − 64 = 0 → x^2 − 4x − 32 = 0.
- Factor: (x − 8)(x + 4) = 0 → x = 8 or x = −4.
- x = 8 gives B; x = −4 gives the other intersection C.
- y = −4 + 8 = 4 → C = (−4, 4).

Algebra tips

Use the distance formula (or sum of squared coordinate differences) to compute r^2 directly.
To find intersections, substitute the line expression for y into the circle (or parametrize x).
Expect two intersection points when a line intersects a circle; identify the known point and take the other as the required vertex.
Solve quadratics by factoring or the quadratic formula; verify both roots geometrically.