Question Statement

The line $12x-5\sqrt{11}y-50=0$ is tangent to the hyperbola $\frac{x^2}{25}-\frac{y^2}{4}=1$.

1. Find the point where the tangent line intersects the focal axis of the hyperbola.
2. Find the ratio in which this point of intersection divides the distance between the foci.

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Background and Explanation

To solve this problem, we need to identify the focal axis of the hyperbola, which is the line passing through its foci. For a horizontal hyperbola of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the focal axis is the $x$-axis ($y=0$). We also use the relationship $c^2=a^2+b^2$ to find the coordinates of the foci and apply the distance formula to determine the ratio of division.

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Solution

Part (a): Finding the Point of Intersection

The given equation of the hyperbola is: $\frac{x^2}{25}-\frac{y^2}{4}=1$

This is a horizontal hyperbola centered at the origin. Its focal axis is the $x$-axis. Therefore, the equation of the focal axis is: $y=0$

The equation of the tangent line is: $12x-5\sqrt{11}y-50=0$

To find the point of intersection $A$ between the tangent line and the focal axis, we substitute $y=0$ from equation (i) into equation (ii): $12x-5\sqrt{11}(0)-50=0$ $12x-50=0$ $12x=50$ $x=\frac{50}{12}=\frac{25}{6}$

Thus, the point of intersection $A$ is: $A\left(\frac{25}{6},0\right)$

Part (b): Finding the Ratio of Division

First, we find the coordinates of the foci. From the hyperbola equation, we have: $a^2=25\text{and}{b}^2=4$

Using the hyperbola identity $b^2=c^2-a^2$: $c^2=a^2+b^2$ $c^2=25+4=29$ $c=\sqrt{29}$

The foci of the hyperbola are $F(c,0)$ and $F^{\prime }(-c,0)$: $F=(\sqrt{29},0)$ $F^{\prime }=(-\sqrt{29},0)$

The distance between the foci is $|F{F}^{\prime }|=2c=2\sqrt{29}$. We now calculate the distances from the intersection point $A$ to each focus to find the ratio.

Distance $|F^{\prime }A|$: $|F^{\prime }A|=\sqrt{{\left(\frac{25}{6}-(-\sqrt{29})\right)}^2+(0-0{)}^2}$ $|F^{\prime }A|=\frac{25}{6}+\sqrt{29}$

Distance $|AF|$: $|AF|=\sqrt{{\left(\sqrt{29}-\frac{25}{6}\right)}^2+(0-0{)}^2}$ $|AF|=\sqrt{29}-\frac{25}{6}$

Calculating the Ratio $|F^{\prime }A|:|AF|$: $\frac{|F^{\prime }A|}{|AF|}=\frac{\frac{25}{6}+\sqrt{29}}{\sqrt{29}-\frac{25}{6}}$

To simplify, multiply the numerator and denominator by $6$: $\frac{|F^{\prime }A|}{|AF|}=\frac{\frac{25+6\sqrt{29}}{6}}{\frac{6\sqrt{29}-25}{6}}$ $\frac{|F^{\prime }A|}{|AF|}=\frac{6\sqrt{29}+25}{6\sqrt{29}-25}$

The required ratio is $(6\sqrt{29}+25):(6\sqrt{29}-25)$.

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Key Formulas or Methods Used

• Standard Hyperbola: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
• Focal Axis: For the given hyperbola, it is the $x$-axis ($y=0$).
• Foci Relation: $c^2=a^2+b^2$, where foci are $(\pm c,0)$.
• Distance Formula: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.
• Ratio of Division: The ratio in which point $A$ divides segment $F^{\prime }F$ is calculated as the ratio of the lengths $|F^{\prime }A|$ and $|AF|$.

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Summary of Steps

1. Identify the focal axis of the hyperbola as $y=0$.
2. Substitute $y=0$ into the tangent line equation to find the intersection point $A(\frac{25}{6},0)$.
3. Calculate the value of $c$ using $c=\sqrt{a^2+b^2}$ to find the foci $F(\sqrt{29},0)$ and $F^{\prime }(-\sqrt{29},0)$.
4. Calculate the distances from the intersection point to both foci ($|F^{\prime }A|$ and $|AF|$).
5. Simplify the ratio of these distances to get the final result.