Question

Consider two circles \( C_1 : x^2 + y^2 = 25 \) and \( C_2 : (x - \alpha)^2 + y^2 = 16 \), where \( \alpha \in (5, 9) \). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of \( C_1 \) and \( C_2 \) be\[\sin^{-1} \left( \frac{\sqrt{63}}{8} \right).\]If the length of the common chord of \( C_1 \) and \( C_2 \) is \( \beta \), then the value of \( (\alpha \beta)^2 \) equals ____.

Answer 1

Correct Answer: 1575

Identify the center and radius of each circle:

- \( C_1 : x^2 + y^2 = 25 \) has center \( (0, 0) \) and radius \( R_1 = 5 \).

- \( C_2 : (x - \alpha)^2 + y^2 = 16 \) has center \( (\alpha, 0) \) and radius \( R_2 = 4 \).

Distance between centers:

\[ d = \alpha \]

Length of the common chord:

\[ \beta = 2 \sqrt{5^2 - \left( \frac{\alpha^2 + 9}{2\alpha} \right)^2} = 2 \sqrt{25 - \left( \frac{\alpha^2 + 9}{2\alpha} \right)^2}. \]

Using \( \sin \theta = \frac{\sqrt{63}}{8} \), calculate \( \alpha \beta \):

\[ \alpha \beta = 5 \sqrt{63} \]

Final calculation: \( (\alpha \beta)^2 = 25 \times 63 =1575 \)

Answer 2

This problem involves the geometry of two intersecting circles. We need to find a relationship between the distance between their centers (\(\alpha\)), the length of their common chord (\(\beta\)), their radii, and the angle between the radii at an intersection point. The final goal is to compute the value of \((\alpha\beta)^2\).

Concept Used:

1. Geometry of Intersecting Circles: When two circles intersect, the centers of the circles and an intersection point form a triangle. Let the centers be \(O_1\) and \(O_2\), the radii be \(r_1\) and \(r_2\), and an intersection point be \(P\). The sides of the triangle \(O_1PO_2\) are \(O_1P = r_1\), \(O_2P = r_2\), and the distance between centers \(O_1O_2 = d\).

2. Area of a Triangle: The area of a triangle can be calculated using two different formulas:

• Using two sides and the included angle: \( \text{Area} = \frac{1}{2}ab\sin\theta \)
• Using base and height: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)

3. Length of Common Chord: The common chord of two intersecting circles is perpendicular to the line segment connecting their centers. Its length is twice the length of the altitude from the intersection point to the line of centers in the triangle \(O_1PO_2\).

Step-by-Step Solution:

Step 1: Identify the parameters of the two circles and the geometric setup.

For circle \(C_1: x^2 + y^2 = 25\):

\[ \text{Center } O_1 = (0, 0), \quad \text{Radius } r_1 = \sqrt{25} = 5 \]

For circle \(C_2: (x - \alpha)^2 + y^2 = 16\):

\[ \text{Center } O_2 = (\alpha, 0), \quad \text{Radius } r_2 = \sqrt{16} = 4 \]

The distance between the centers is \(d = \sqrt{(\alpha - 0)^2 + (0 - 0)^2} = \alpha\).

Let \(P\) be one of the intersection points. This forms a triangle \(\triangle O_1PO_2\) with side lengths \(O_1P = r_1 = 5\), \(O_2P = r_2 = 4\), and \(O_1O_2 = \alpha\). The angle between the radii at point \(P\) is \(\angle O_1PO_2 = \theta\), where it is given that:

\[ \theta = \sin^{-1} \left( \frac{\sqrt{63}}{8} \right) \implies \sin\theta = \frac{\sqrt{63}}{8} \]

Step 2: Relate the length of the common chord, \(\beta\), to the area of \(\triangle O_1PO_2\).

The common chord is perpendicular to the line of centers \(O_1O_2\). The length of the common chord, \(\beta\), is twice the length of the altitude from vertex \(P\) to the base \(O_1O_2\) of \(\triangle O_1PO_2\). Let this altitude be \(h\). Then \(\beta = 2h\).

The area of \(\triangle O_1PO_2\) can be calculated in two ways:

Method A (using base and height):

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} (O_1O_2)(h) = \frac{1}{2} \alpha \left(\frac{\beta}{2}\right) = \frac{\alpha\beta}{4} \]

Method B (using the angle between two sides):

\[ \text{Area} = \frac{1}{2} (O_1P)(O_2P) \sin(\angle O_1PO_2) = \frac{1}{2} r_1 r_2 \sin\theta \]

Step 3: Equate the two expressions for the area to find a formula for the product \(\alpha\beta\).

By equating the two expressions for the area, we get:

\[ \frac{\alpha\beta}{4} = \frac{1}{2} r_1 r_2 \sin\theta \]

Substitute the values \(r_1 = 5\) and \(r_2 = 4\):

\[ \frac{\alpha\beta}{4} = \frac{1}{2} (5)(4) \sin\theta = 10 \sin\theta \]

Solving for \(\alpha\beta\):

\[ \alpha\beta = 40 \sin\theta \]

Step 4: Substitute the given value of \(\sin\theta\) into the expression for \(\alpha\beta\).

We are given that \( \sin\theta = \frac{\sqrt{63}}{8} \). Substituting this into our equation:

\[ \alpha\beta = 40 \left( \frac{\sqrt{63}}{8} \right) \] \[ \alpha\beta = 5\sqrt{63} \]

Final Computation & Result:

The problem asks for the value of \((\alpha\beta)^2\). We square the result from Step 4:

\[ (\alpha\beta)^2 = (5\sqrt{63})^2 \] \[ = 5^2 \times (\sqrt{63})^2 \] \[ = 25 \times 63 \]

To compute the product:

\[ 25 \times 63 = 25 \times (60 + 3) = 25 \times 60 + 25 \times 3 = 1500 + 75 = 1575 \]

The value of \( (\alpha \beta)^2 \) is 1575.