The curve, y=x^2 -6x +5, and the circle, (x-3)^2 +(y-3/2)^2 =85/2, intersect at the points A(2,-3) B(6,5) and C(0,5). Find the fourth point of intersection

To find the fourth point of intersection between the curve \( y = x^2 - 6x + 5 \) and the circle \( (x - 3)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{85}{2} \), we can use the fact that if points \( A(2, -3), B(6, 5), \) and \( C(0, 5) \) are already intersection points, substituting these into both equations should satisfy them. We will solve for any additional solutions beyond these known points.

### Step 1: Substitute \( y = x^2 - 6x + 5 \) into the Circle Equation
Given:
\[
y = x^2 - 6x + 5
\]
Substitute this into the circle equation:
\[
(x - 3)^2 + \left((x^2 - 6x + 5) - \frac{3}{2}\right)^2 = \frac{85}{2}
\]

### Step 2: Simplify and Solve for \( x \)
Expanding and solving this equation will yield the \( x \)-coordinates for all intersection points. Since we already know three points, this will result in a fourth \( x \)-coordinate.

### Step 3: Find Corresponding \( y \)-Coordinate
Once we have the fourth \( x \)-coordinate, substitute it back into \( y = x^2 - 6x + 5 \) to find the corresponding \( y \)-coordinate.

Let me perform the calculations for you.The fourth point of intersection for the given curve and circle, after excluding the known intersection points, is approximately:

\[
\left( -0.33, \, 7.10 \right)
\]

Would you like further details on any of the calculation steps?

Here are some related questions for further exploration:

1. How would you verify this intersection point graphically?
2. What is the significance of finding intersection points between a curve and a circle?
3. How could you confirm that this fourth point satisfies both the circle and parabola equations?
4. Could there be any real-life applications of finding intersection points between two shapes?
5. How would this problem change if the equations represented a line and a circle?

**Tip:** To check calculations involving intersections, verify each point by substituting back into the original equations.

The curve, y=x^2 -6x +5, and the circle, (x-3)^2 +(y-3/2)^2 =85/2, intersect at the points A(2,-3) B(6,5) and C(0,5). Find the fourth point of intersection.

Learn how to find the fourth intersection point between the curve y = x^2 - 6x + 5 and the circle (x - 3)^2 + (y - 3/2)^2 = 85/2. This solution covers key concepts in analytic geometry, including circle and parabola equations, and involves solving quadratic equations. Suitable for students in Grades 10-12.

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