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Animated Solution for Mathematics - Circles: If the two circles $(x - 1)^{2} + (y - 3)^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct points, then

Q: Wait, why do we use the absolute difference $|r_1 - r_2|$ in the condition?

Great question! The condition $|r_1 - r_2| |r_1 - r_2|$, ensures that one circle is not completely contained within the other. If $d = |r_1 - r_2|$, the circles touch internally, which is not an intersection at two distinct points.

Q: Why can't we just use energy conservation or some other physics principle here?

This is a pure coordinate geometry problem, not a mechanics problem! While physics often uses energy conservation, here we are dealing with the spatial relationship between geometric shapes. We rely on the properties of circles and the distance formula to define their intersection.

Q: Is it possible for $r$ to be negative?

No, $r$ represents the radius of a circle, which must always be a positive real number. When solving the inequality $|r - 3| 2$, we get the final range $2 < r < 8$.

Select Answer:

Progressive Hints

Hint 1

Hint 1: Visualize the two circles on the coordinate plane. The key to their intersection lies in the relationship between the distance between their centers,

d

, and their radii,

r_{1}

and

r_{2}

. What geometric condition must be satisfied for two circles to intersect at exactly two distinct points?

Hint 2

Hint 2: Recall the triangle inequality for circles. For two circles to intersect at two distinct points, the distance between their centers

d

must satisfy the inequality

∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}

. This ensures the circles are neither too far apart nor one inside the other.

Hint 3

Hint 3: First, find the centers and radii of both circles by completing the square. Then, calculate the distance

d

between the centers

(1, 3)

and

(4, - 1)

. Finally, substitute

r_{1} = r

r_{2} = 3

, and

d = 5

into the inequality

∣ r - 3∣ < 5 < r + 3

and solve for

r

Visualized Solution

Visualizing the Problem

Given circles:
$C_{1} : (x - 1)^{2} + (y - 3)^{2} = r^{2}$
$C_{2} : x^{2} + y^{2} - 8 x + 2 y + 8 = 0$
Goal: Find the range of $r$ for two distinct intersection points.

Analyzing Circle 1

First circle equation: $(x - 1)^{2} + (y - 3)^{2} = r^{2}$
Standard form: $(x - h)^{2} + (y - k)^{2} = R^{2}$
Center $C_{1} = (1, 3)$
Radius $r_{1} = r$

Analyzing Circle 2

Second circle: $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$
Grouping terms: $(x^{2} - 8 x) + (y^{2} + 2 y) + 8 = 0$
Completing the square for $x$ and $y$ terms.

Circle 2 Center and Radius

$(x - 4)^{2} - 16 + (y + 1)^{2} - 1 + 8 = 0$
$(x - 4)^{2} + (y + 1)^{2} = 9$
$(x - 4)^{2} + (y + 1)^{2} = 3^{2}$
Center $C_{2} = (4, - 1)$ , Radius $r_{2} = 3$

Distance Between Centers

Distance $d = C_{1} C_{2}$
Using distance formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Substitute: $d = (4 - 1)^{2} + (- 1 - 3)^{2}$

Computing the Distance

$d = 3^{2} + (- 4)^{2}$
$d = 9 + 16$
$d = 25 = 5$

The Intersection Condition

Condition for two distinct intersection points:
$∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}$
This ensures the circles neither completely enclose each other nor stay completely apart.

Substituting the Values

We know: $r_{1} = r$ , $r_{2} = 3$ , $d = 5$
Substitute into the condition:
$∣ r - 3∣ < 5 < r + 3$

Solving the Right Inequality

Part 1: $5 < r + 3$
Subtract $3$ from both sides:
$5 - 3 < r$
$r > 2$

Solving the Left Inequality

Part 2: $∣ r - 3∣ < 5$
Expanding the modulus:
$- 5 < r - 3 < 5$

Finalizing the Range

$- 5 < r - 3 < 5$
Add $3$ to all parts:
$- 5 + 3 < r < 5 + 3$
$- 2 < r < 8$

The Final Conclusion

Condition 1: $r > 2$
Condition 2: $- 2 < r < 8$
Taking the intersection of both conditions:
$2 < r < 8$

00:00 / 00:00

The Sigma Insight: Standard and General Equation of a Circle

Solution Diagram

Analyzing the Setup

We are tasked with finding the range of the radius

r

for the circle

C_{1}

such that it intersects the circle

C_{2}

at exactly two distinct points.

The first circle is defined by the equation:

(x - 1)^{2} + (y - 3)^{2} = r^{2}

By inspection, the center of

C_{1}

O_{1} = (1, 3)

and its radius is

r_{1} = r

Decoding the Second Circle

The second circle is given by the general equation:

x^{2} + y^{2} - 8 x + 2 y + 8 = 0

To identify its properties, we complete the square for both

x

and

y

variables:

(x^{2} - 8 x + 16) + (y^{2} + 2 y + 1) = - 8 + 16 + 1

This simplifies to the standard form:

(x - 4)^{2} + (y + 1)^{2} = 9

Thus, the center of

C_{2}

O_{2} = (4, - 1)

and its radius is

r_{2} = 9 = 3

Calculating the Distance Between Centers

We determine the distance

d

between the centers

O_{1} (1, 3)

and

O_{2} (4, - 1)

using the distance formula:

d = (4 - 1)^{2} + (- 1 - 3)^{2}

d = 3^{2} + (- 4)^{2} = 9 + 16 = 25 = 5

The Condition for Intersection

For two circles to intersect at exactly two distinct points, the distance between their centers must satisfy the triangle inequality relative to their radii:

∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}

Substituting our known values

r_{1} = r

r_{2} = 3

, and

d = 5

, we obtain:

∣ r - 3∣ < 5 < r + 3

Final Calculation

We solve this compound inequality in two parts:

1. From

5 < r + 3

, we find:

r > 2

2. From

∣ r - 3∣ < 5

, we expand the absolute value:

- 5 < r - 3 < 5

- 2 < r < 8

Combining these two conditions (

r > 2

and

- 2 < r < 8

), we arrive at the final range for

r

: $2 < r < 8$

Common Traps & FAQs

Wait, why do we use the absolute difference

∣ r_{1} - r_{2} ∣

in the condition?

Great question! The condition

∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}

is essential because it covers both boundaries. The right side,

d < r_{1} + r_{2}

, ensures the circles are close enough to touch or intersect. The left side,

d > ∣ r_{1} - r_{2} ∣

, ensures that one circle is not completely contained within the other. If

d = ∣ r_{1} - r_{2} ∣

, the circles touch internally, which is not an intersection at two distinct points.

Why can't we just use energy conservation or some other physics principle here?

This is a pure coordinate geometry problem, not a mechanics problem! While physics often uses energy conservation, here we are dealing with the spatial relationship between geometric shapes. We rely on the properties of circles and the distance formula to define their intersection.

Is it possible for

r

to be negative?

No,

r

represents the radius of a circle, which must always be a positive real number. When solving the inequality

∣ r - 3∣ < 5

, we get

- 2 < r < 8

. However, since

r

must be positive, we only consider the range

0 < r < 8

. Combined with our other condition

r > 2

, we get the final range

2 < r < 8

Animated Solution for Mathematics - Circles: If the two circles $(x - 1)^{2} + (y - 3)^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct points, then

Select Answer:

Visualized Solution

Visualizing the Problem

Analyzing Circle 1

Analyzing Circle 2

Circle 2 Center and Radius

Distance Between Centers

Computing the Distance

The Intersection Condition

Substituting the Values

Solving the Right Inequality

Solving the Left Inequality

Finalizing the Range

The Final Conclusion

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

Decoding the Second Circle

Calculating the Distance Between Centers

The Condition for Intersection

Final Calculation

Similar Questions

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Animated Solution for Mathematics - Circles: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the two circles (x−1)2+(y−3)2=r2 and x2+y2−8x+2y+8=0 intersect in two distinct points, then

Select Answer:

Visualized Solution

Visualizing the Problem

Analyzing Circle 1

Analyzing Circle 2

Circle 2 Center and Radius

Distance Between Centers

Computing the Distance

The Intersection Condition

Substituting the Values

Solving the Right Inequality

Solving the Left Inequality

Finalizing the Range

The Final Conclusion

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

Decoding the Second Circle

Calculating the Distance Between Centers

The Condition for Intersection

Final Calculation

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