Question

Consider a hyperbola \( H \) having its centre at the origin and foci on the \( x \)-axis. Let \( C_1 \) be the circle touching the hyperbola \( H \) and having its centre at the origin. Let \( C_2 \) be the circle touching the hyperbola \( H \) at its vertex and having its centre at one of its foci. If the areas (in square units) of \( C_1 \) and \( C_2 \) are \( 36\pi \) and \( 4\pi \), respectively, then the length (in units) of the latus rectum of \( H \) is:

• A. \( \frac{28}{3} \)
• B. \( \frac{14}{3} \)
• C. \( \frac{10}{3} \)
• D. \( \frac{11}{3} \)

Answer 1

The Correct Option is A

The area of a circle is given by \( A = \pi r^2 \).

1. For circle \( C_1 \), we are given that \( A = 36\pi \). Therefore, the radius \( r_1 \) of \( C_1 \) is: \[ \pi r_1^2 = 36\pi \implies r_1 = 6. \]
2. For circle \( C_2 \), we are given that \( A = 4\pi \). Therefore, the radius \( r_2 \) of \( C_2 \) is: \[ \pi r_2^2 = 4\pi \implies r_2 = 2. \]

Since \( C_1 \) is centered at the origin and touches the hyperbola, the radius \( r_1 = 6 \) is equal to the distance from the center to the vertex of the hyperbola, which is \( a \) (the semi-major axis length). Therefore, \( a = 6 \).

For the hyperbola centered at the origin with foci along the \( x \)-axis, the focal distance \( c \) is the distance from the origin to one of the foci. Since \( C_2 \) has its center at one of the foci and radius \( r_2 = 2 \), we find that \( c - a = 2 \). Thus, \[ c = a + 2 = 6 + 2 = 8. \]

The relationship between \( a \), \( b \), and \( c \) for a hyperbola is given by \( c^2 = a^2 + b^2 \). Substituting the known values: \[ 8^2 = 6^2 + b^2 \implies 64 = 36 + b^2 \implies b^2 = 28 \implies b = \sqrt{28}. \]

The length of the latus rectum for a hyperbola is given by \( \frac{2b^2}{a} \). Therefore, the length of the latus rectum is: \[ \frac{2b^2}{a} = \frac{2 \times 28}{6} = \frac{56}{6} = \frac{28}{3}. \]

Thus, the length of the latus rectum of \( H \) is \( \frac{28}{3} \).

Answer 2

Step 1: Understand the given information.
We are given a hyperbola \( H \) centered at the origin, with its foci on the \( x \)-axis. Two circles are mentioned:
- \( C_1 \), a circle centered at the origin that touches the hyperbola \( H \),
- \( C_2 \), a circle centered at one of the foci that touches the hyperbola at its vertex.
The areas of the circles are given as:
- Area of \( C_1 = 36\pi \),
- Area of \( C_2 = 4\pi \).
We need to find the length of the latus rectum of the hyperbola \( H \).

Step 2: Relating the areas of the circles to the radii.
The area of a circle is given by \( \pi r^2 \), where \( r \) is the radius. From the given areas:
- For \( C_1 \):
\[ \pi r_1^2 = 36\pi \quad \Rightarrow \quad r_1^2 = 36 \quad \Rightarrow \quad r_1 = 6. \] - For \( C_2 \):
\[ \pi r_2^2 = 4\pi \quad \Rightarrow \quad r_2^2 = 4 \quad \Rightarrow \quad r_2 = 2. \] So, the radii of the circles are \( r_1 = 6 \) and \( r_2 = 2 \).

Step 3: Equation of the hyperbola and the relationship between the radii.
The general equation of the hyperbola with its center at the origin and foci along the \( x \)-axis is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \] where \( 2a \) is the distance between the vertices and \( 2c \) is the distance between the foci, with \( c^2 = a^2 + b^2 \). The length of the latus rectum is given by \( \frac{2b^2}{a} \).
The circle \( C_1 \) touches the hyperbola at the origin, and its radius is \( r_1 = 6 \), which is the distance from the origin to the vertex of the hyperbola. Therefore, we have:
\[ a = 6. \] The circle \( C_2 \) touches the hyperbola at the vertex and is centered at one of the foci. Its radius is \( r_2 = 2 \), and it is tangent to the hyperbola at the vertex. The distance from the focus to the vertex is \( c - a \), so:
\[ c - a = r_2 = 2. \] Since \( a = 6 \), we find:
\[ c = a + 2 = 6 + 2 = 8. \] Now, we can use the relationship \( c^2 = a^2 + b^2 \) to find \( b^2 \):
\[ c^2 = a^2 + b^2 \quad \Rightarrow \quad 8^2 = 6^2 + b^2 \quad \Rightarrow \quad 64 = 36 + b^2 \quad \Rightarrow \quad b^2 = 28. \]

Step 4: Find the length of the latus rectum.
The length of the latus rectum of the hyperbola is given by \( \frac{2b^2}{a} \). Substituting the values of \( b^2 = 28 \) and \( a = 6 \):
\[ \text{Length of the latus rectum} = \frac{2b^2}{a} = \frac{2 \times 28}{6} = \frac{56}{6} = \frac{28}{3}. \]

Step 5: Conclusion.
Thus, the length of the latus rectum of the hyperbola is \( \boxed{\frac{28}{3}} \).