Understanding Circle Intersection

The question asks for the condition on the radius \(r\) of the first circle such that it intersects the second given circle at two distinct points. To solve this, we need to analyze the properties of both circles, specifically their centers and radii, and then use the condition for two circles to intersect at two distinct points based on the distance between their centers.

Analysis of the First Circle

The equation of the first circle is given as \(x^2 + y^2 = r^2\).

• This equation is in the standard form \((x-h)^2 + (y-k)^2 = R^2\), where \((h, k)\) is the center and \(R\) is the radius.
• Comparing \(x^2 + y^2 = r^2\) with the standard form, we can see that the center of the first circle, let's call it \(C_1\), is at \((0, 0)\).
• The radius of the first circle, let's call it \(R_1\), is \(r\). Since \(r\) represents a radius, we know that \(r > 0\).

Analysis of the Second Circle

The equation of the second circle is given as \(x^2 + y^2 – 10x + 16 = 0\).

• This equation is in the general form \(x^2 + y^2 + 2gx + 2fy + c = 0\).
• Comparing \(x^2 + y^2 – 10x + 16 = 0\) with the general form, we have \(2g = -10\), \(2f = 0\), and \(c = 16\).
• From this, we find \(g = -5\) and \(f = 0\).
• The center of the second circle, let's call it \(C_2\), is at \((-g, -f)\), which is \((-(-5), -0)) = (5, 0)\).
• The radius of the second circle, let's call it \(R_2\), is calculated using the formula \(R_2 = \sqrt{g^2 + f^2 - c}\).
• Substituting the values, \(R_2 = \sqrt{(-5)^2 + 0^2 - 16} = \sqrt{25 + 0 - 16} = \sqrt{9} = 3\).

Calculating the Distance Between Centers

The distance between the centers \(C_1 = (0, 0)\) and \(C_2 = (5, 0)\), let's call it \(d\), is calculated using the distance formula:

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

\(d = \sqrt{(5 - 0)^2 + (0 - 0)^2}\)

\(d = \sqrt{5^2 + 0^2}\)

\(d = \sqrt{25}\)

\(d = 5\)

The distance between the centers is 5 units.

Condition for Two Distinct Intersection Points

Two circles intersect at two distinct points if and only if the distance between their centers is strictly greater than the absolute difference of their radii and strictly less than the sum of their radii.

Mathematically, the condition is: \(|R_1 - R_2| < d < R_1 + R_2\)

Substituting the values \(R_1 = r\), \(R_2 = 3\), and \(d = 5\):

\(|r - 3| < 5 < r + 3\)

Solving the Inequality for r

We need to solve the compound inequality \(|r - 3| < 5\) and \(5 < r + 3\) simultaneously.

Part 1: \(|r - 3| < 5\)

This inequality is equivalent to:

\(-5 < r - 3 < 5\)

Add 3 to all parts of the inequality:

\(-5 + 3 < r - 3 + 3 < 5 + 3\)

\(-2 < r < 8\)

Since \(r\) is a radius, it must be a positive value. Therefore, considering \(r > 0\), this part of the condition gives \(0 < r < 8\).

Part 2: \(5 < r + 3\)

Subtract 3 from both sides of the inequality:

\(5 - 3 < r + 3 - 3\)

\(2 < r\)

Combining the Conditions

For the circles to intersect at two distinct points, both \(0 < r < 8\) and \(2 < r\) must be true. The intersection of these two conditions is \(2 < r < 8\).

This means the radius \(r\) of the first circle must be greater than 2 and less than 8 for the two circles to intersect at two distinct points.

Revision Table: Circle Intersection Conditions

Here is a summary of the conditions for the intersection of two circles with radii \(R_1\), \(R_2\) and distance between centers \(d\).

Additional Information: General Circle Equation

The general equation of a circle is \(x^2 + y^2 + 2gx + 2fy + c = 0\). From this equation, we can find the center and radius:

• The center of the circle is at the point \((-g, -f)\).
• The radius of the circle is \(R = \sqrt{g^2 + f^2 - c}\).
• For the equation to represent a real circle, the term under the square root must be non-negative: \(g^2 + f^2 - c \ge 0\).
• If \(g^2 + f^2 - c > 0\), it is a real circle with a positive radius.
• If \(g^2 + f^2 - c = 0\), it represents a point circle (radius is 0).
• If \(g^2 + f^2 - c < 0\), it represents an imaginary circle.

In our problem, for the second circle \(x^2 + y^2 – 10x + 16 = 0\), we found \(g=-5\), \(f=0\), and \(c=16\). So \(g^2+f^2-c = (-5)^2 + 0^2 - 16 = 25 - 16 = 9 > 0\), confirming it is a real circle with radius 3.