Question

The length of the latus rectum and directrices of a hyperbola with eccentricity $e$ are 9 and $x = \pm \frac{4}{\sqrt{3}}$, respectively. Let the line $y - \sqrt{3}x + \sqrt{3} = 0$ touch this hyperbola at $(x_0, y_0)$. If $m$ is the product of the focal distances of the point $(x_0, y_0)$, then $4e^2 + m$ is equal to ________.

Answer 1

Correct Answer: 61

The objective is to determine the value of \(4e^2 + m\). Here, \(e\) represents the eccentricity of a hyperbola, and \(m\) is the product of the focal distances for a point of tangency \((x_0, y_0)\). The provided information includes the hyperbola's latus rectum, the equations of its directrices, and the equation of a tangent line.

Concept Used:

The solution is based on the standard properties of a hyperbola with a horizontal transverse axis, defined by the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

1. Eccentricity Relation: \(b^2 = a^2(e^2 - 1)\).
2. Length of Latus Rectum: \( \frac{2b^2}{a} \).
3. Equations of Directrices: \( x = \pm \frac{a}{e} \).
4. Condition of Tangency: A line \(y = mx + c\) is tangent to the hyperbola if \(c^2 = a^2m^2 - b^2\).
5. Equation of Tangent: The tangent to the hyperbola at point \((x_0, y_0)\) is \(\frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1\).
6. Focal Distances: For a point \(P(x_0, y_0)\) on the hyperbola, the distances from foci \(S(ae, 0)\) and \(S'(-ae, 0)\) are \(|ex_0 - a|\) and \(|ex_0 + a|\).
7. Product of Focal Distances: The product is \(m = |(ex_0 - a)(ex_0 + a)| = |e^2x_0^2 - a^2|\).

Note: An inconsistency exists in the problem statement as the latus rectum, directrix, and tangent line properties provided do not correspond to the same hyperbola. The hyperbola's parameters will be derived using the latus rectum and the tangent line condition, as they form a consistent system yielding a unique solution.

Step-by-Step Solution:

Step 1: Formulate equations from the given information.

Given latus rectum length is 9:

\[\frac{2b^2}{a} = 9 \implies b^2 = \frac{9a}{2} \quad \text{(Equation 1)}\]

The tangent line is \(y - \sqrt{3}x + \sqrt{3} = 0\), re-arranged as \(y = \sqrt{3}x - \sqrt{3}\). The slope is \(m_{\text{tan}} = \sqrt{3}\) and the y-intercept is \(c = -\sqrt{3}\).

Applying the tangency condition, \(c^2 = a^2m_{\text{tan}}^2 - b^2\):

\[(-\sqrt{3})^2 = a^2(\sqrt{3})^2 - b^2\]\[3 = 3a^2 - b^2 \quad \text{(Equation 2)}\]

Step 2: Solve for the hyperbola's parameters \(a\) and \(b\).

Substitute Equation 1 into Equation 2:

\[3 = 3a^2 - \frac{9a}{2}\]

Multiply by 2 to clear the fraction:

\[6 = 6a^2 - 9a\]

Form a quadratic equation:

\[6a^2 - 9a - 6 = 0\]

Divide by 3:

\[2a^2 - 3a - 2 = 0\]

Factor the quadratic equation:

\[(2a + 1)(a - 2) = 0\]

Since \(a\), the semi-major axis length, must be positive, \(a = 2\).

Calculate \(b^2\) using Equation 1:

\[b^2 = \frac{9(2)}{2} = 9\]

The hyperbola's equation is \(\frac{x^2}{4} - \frac{y^2}{9} = 1\).

Step 3: Calculate \(e^2\), the square of the eccentricity.

Using the relation \(b^2 = a^2(e^2 - 1)\):

\[9 = 4(e^2 - 1)\]\[\frac{9}{4} = e^2 - 1\]\[e^2 = \frac{9}{4} + 1 = \frac{13}{4}\]

Step 4: Determine the point of tangency \((x_0, y_0)\).

The tangent equation at \((x_0, y_0)\) is \(\frac{xx_0}{4} - \frac{yy_0}{9} = 1\). The given tangent line is \(\sqrt{3}x - y = \sqrt{3}\). Comparing coefficients:

\[\frac{x_0/4}{\sqrt{3}} = \frac{-y_0/9}{-1} = \frac{1}{\sqrt{3}}\]

From \(\frac{x_0}{4\sqrt{3}} = \frac{1}{\sqrt{3}}\), we get \(x_0 = 4\).

From \(\frac{y_0}{9} = \frac{1}{\sqrt{3}}\), we get \(y_0 = \frac{9}{\sqrt{3}} = 3\sqrt{3}\).

The point of tangency is \((x_0, y_0) = (4, 3\sqrt{3})\).

Step 5: Calculate \(m\), the product of the focal distances.

The formula for the product of focal distances is \(m = |e^2x_0^2 - a^2|\).

Substitute \(e^2 = \frac{13}{4}\), \(x_0 = 4\), and \(a = 2\):

\[m = \left| \left(\frac{13}{4}\right)(4)^2 - (2)^2 \right|\]\[m = \left| \left(\frac{13}{4}\right)(16) - 4 \right|\]\[m = |13 \times 4 - 4| = |52 - 4| = 48\]

Step 6: Compute the final value of \(4e^2 + m\).

Substitute the values of \(e^2\) and \(m\):

\[4e^2 + m = 4\left(\frac{13}{4}\right) + 48\]\[= 13 + 48 = 61\]

The value of \(4e^2 + m\) is 61.