Let A(α, 0) and B(0, β) be the points on the line 5x + 7y = 50. Let the point P divide the line segm

Q: Wait, why is the perpendicular from the focus to the x-axis just $x = ae$? Isn't the focus a point?

Great question! The focus $S$ of a standard ellipse is at $(ae, 0)$. A line perpendicular to the x-axis is a vertical line. Since this vertical line must pass through $(ae, 0)$, its equation is simply $x = ae$. It is a vertical slice through the focus, and every point on that line shares the same x-coordinate!

Q: Why did we multiply the two equations $\frac{a}{e} = \frac{25}{3}$ and $ae = 3$ instead of dividing them?

You could divide them, which would give you $e^2 = \frac{9}{25}$, and that is perfectly valid! However, multiplying them is a beautiful algebraic shortcut. It immediately eliminates $e$ and gives you $a^2$ in one step. Always look for the path of least resistance in your algebra to save time during the exam.

Q: Is it possible for the ellipse to be vertical, i.e., $b > a$?

In this problem, the directrix is given as $x = \frac{25}{3}$, which is a vertical line. This implies the major axis is along the x-axis. If the major axis were along the y-axis, the directrix would be a horizontal line $y = \frac{b}{e}$. Since we are given $x = \text{constant}$, we know the major axis is horizontal.

Question

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x + 7y = 50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7 : 3$. Let $3x - 25 = 0$ be a directrix of the ellipse $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to

Options:

(A) $\frac{25}{3}$

(B) $\frac{32}{9}$

(C) $\frac{25}{9}$

(D) $\frac{32}{5}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We begin with the line equation $5x + 7y = 50$. This line serves as our geometric anchor.

To define the segment $AB$, we calculate the intercepts. Setting $y=0$ yields $A(10, 0)$, and setting $x=0$ yields $B(0, \frac{50}{7})$.

Point $P$ lies on segment $AB$ and divides it in a ratio of $7:3$. Applying the section formula, we determine the coordinates of $P$:

$$P = \left( \frac{7(0) + 3(10)}{7+3}, \frac{7(\frac{50}{7}) + 3(0)}{7+3} ight) = (3, 5)$$

This point $P(3, 5)$ is our fixed reference point for the ellipse.

### The Ellipse Mystery

We are given the directrix of the ellipse as $x = \frac{25}{3}$. For an ellipse in the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the directrix is defined by the equation $x = \frac{a}{e}$.

This provides our first relationship:

$$\frac{a}{e} = \frac{25}{3}$$

The focus $S$ of this ellipse is located at $(ae, 0)$. A line perpendicular to the x-axis passing through the focus is $x = ae$. Since this line must pass through our fixed point $P(3, 5)$, the x-coordinate of the focus must be $3$.

Thus, we establish our second relationship:

$$ae = 3$$

### The Algebraic Elegance

We now solve the system of equations involving $a$ and $e$. By multiplying the two equations, we eliminate the eccentricity $e$:

$$\left( \frac{a}{e} ight) \cdot (ae) = \left( \frac{25}{3} ight) \cdot 3$$

$$a^2 = 25 \Rightarrow a = 5$$

Substituting $a=5$ back into $ae = 3$, we find the eccentricity:

$$e = \frac{3}{5}$$

### Final Calculation

To determine the length of the latus rectum, we first calculate $b^2$ using the fundamental ellipse identity $b^2 = a^2(1 - e^2)$:

$$b^2 = 25 \left( 1 - \left( \frac{3}{5} ight)^2 ight) = 25 \left( 1 - \frac{9}{25} ight) = 25 \left( \frac{16}{25} ight) = 16$$

The length of the latus rectum is defined by the formula $\frac{2b^2}{a}$. Substituting our derived values:

$$	ext{Length} = \frac{2(16)}{5} = \frac{32}{5}$$

The final length of the latus rectum is **$\frac{32}{5}$**.

Select Answer:

Visualized Solution

Visualizing the Line $5 x + 7 y = 50$

Finding Point $A$

Finding Point $B$

Section Formula for Point $P$

Calculating Coordinates of $P$

Analyzing Ellipse Directrix

Focus $S$ and the Vertical Line

Relating Focus to Point $P$

Solving for $a^{2}$

Finding $a$ and $e$

Calculating $b^{2}$

Final Latus Rectum Calculation

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

The Ellipse Mystery

The Algebraic Elegance

Final Calculation

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Visualized Solution

Visualizing the Line .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 5x+7y=50

Finding Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Finding Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B

Section Formula for Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P

Calculating Coordinates of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P