- Translate the problem requirements: We need to find how many integer values of radius r are possible given that point \((5,7)\) must be outside the circle and point \((1,-2)\) must be inside the circle, where the center is at \((5,-2)\).
- Calculate boundary distances: Find the exact distances from the center to both given points to establish the constraints on the radius.
- Apply circle position rules: Use the fact that for a point to be inside, its distance from center must be less than r, and for a point to be outside, its distance must be greater than r.
- Identify valid integer radius range: Determine which integer values of r satisfy both constraints simultaneously and count them.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in plain English:
- We have a circle with its center at point \((5, -2)\)
- Point \((5, 7)\) is outside the circle - this means it's farther from the center than the radius
- Point \((1, -2)\) is inside the circle - this means it's closer to the center than the radius
- The radius r must be a whole number (integer)
We need to find how many different integer values the radius could have while satisfying both conditions.
Process Skill: TRANSLATE - Converting the geometric relationships into mathematical constraints
2. Calculate boundary distances
To solve this, we need to find how far each point is from the center of the circle.
Distance from center \((5, -2)\) to the outside point \((5, 7)\):
Think of this as finding the length of a straight line between these two points.
From \((5, -2)\) to \((5, 7)\): we move 0 units horizontally and 9 units vertically
Distance = \(\sqrt{(5-5)^2 + (7-(-2))^2} = \sqrt{0^2 + 9^2} = \sqrt{81} = 9\)
Distance from center \((5, -2)\) to the inside point \((1, -2)\):
From \((5, -2)\) to \((1, -2)\): we move 4 units horizontally and 0 units vertically
Distance = \(\sqrt{(5-1)^2 + (-2-(-2))^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4\)
3. Apply circle position rules
Now we use the fundamental rules about circles:
- For a point to be inside a circle: its distance from center < radius
- For a point to be outside a circle: its distance from center > radius
Applying this to our situation:
- Since \((1, -2)\) is inside: \(4 < r\) (the radius must be greater than 4)
- Since \((5, 7)\) is outside: \(9 > r\) (the radius must be less than 9)
Combining both conditions: \(4 < r < 9\)
Process Skill: APPLY CONSTRAINTS - Using the geometric rules to create mathematical inequalities
4. Identify valid integer radius range
We need integer values of r where \(4 < r < 9\).
The integers between 4 and 9 (not including 4 and 9) are:
- \(r = 5\) ✓
- \(r = 6\) ✓
- \(r = 7\) ✓
- \(r = 8\) ✓
Let's verify one example: If \(r = 6\)
- Point \((1, -2)\) at distance 4 from center: \(4 < 6\) ✓ (inside)
- Point \((5, 7)\) at distance 9 from center: \(9 > 6\) ✓ (outside)
Therefore, there are exactly 4 possible integer values for the radius.
Final Answer
The answer is (A) 4
The four possible integer radius values are 5, 6, 7, and 8, each satisfying the constraint that \(4 < r < 9\).
Common Faltering Points
Errors while devising the approach
- Misinterpreting "inside" and "outside" conditions: Students may incorrectly think that if a point is "inside" the circle, then distance from center \(\leq\) radius, and if "outside" then distance \(\geq\) radius. The correct interpretation is that "inside" means distance < radius (strictly less than) and "outside" means distance > radius (strictly greater than). This error leads to including boundary values incorrectly.
- Confusing which inequality direction to use: Students may set up the constraints backwards, thinking that since \((5,7)\) is outside, we need \(r > 9\), or since \((1,-2)\) is inside, we need \(r < 4\). The correct setup requires understanding that the radius must be larger than the inside point's distance but smaller than the outside point's distance.
- Not recognizing this as a constraint problem: Students might try to find a single "correct" radius value instead of understanding that we need to find all integer values that satisfy both the inside and outside conditions simultaneously.
Errors while executing the approach
- Distance formula calculation errors: Students commonly make arithmetic mistakes when applying the distance formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). For example, calculating \((7-(-2))\) as 5 instead of 9, or forgetting to square the coordinate differences before adding them.
- Sign errors with negative coordinates: When working with the center point \((5, -2)\), students may incorrectly handle the negative y-coordinate, calculating \((-2-(-2))\) incorrectly or making errors with \((7-(-2))\).
- Incorrectly combining the inequalities: Students may write the constraint as \(4 > r > 9\) instead of \(4 < r < 9\), or may solve each inequality separately without properly combining them into a single range.
Errors while selecting the answer
- Including boundary values in the count: Students may incorrectly include \(r = 4\) or \(r = 9\) in their final count, thinking the inequality allows for equality. Since we need \(4 < r < 9\) (strict inequalities), only 5, 6, 7, and 8 are valid.
- Miscounting the integers in the range: Students may incorrectly count the integers between 4 and 9, perhaps listing {4, 5, 6, 7, 8, 9} and getting 6 values, or missing one of the valid integers and getting 3 values instead of 4.
