Key Concepts and Formulas

• Distance of a Point from a Line: The distance $D$ of a point $(x_1, y_1)$ from a line $Ax + By + C = 0$ is given by $D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.
• Equation of a Parabola: A parabola with axis parallel to the y-axis has the form $(x - h)^2 = 4a(y - k)$, where $(h, k)$ is the vertex and $4a$ is the length of the latus rectum.
• Differential Equations: To form a differential equation, differentiate the general equation repeatedly to eliminate arbitrary constants.

Step-by-Step Solution

Step 1: Calculate the Length of the Latus Rectum

We are given the point $(2, -3)$ and the line $3x + 4y = 5$ or $3x + 4y - 5 = 0$. We need to find the distance between them, which will be the length of the latus rectum.

Why this step first? The length of the latus rectum is a fixed value for all parabolas in the family. Calculating it upfront simplifies the general equation.

Using the distance formula: $L = \frac{|3(2) + 4(-3) - 5|}{\sqrt{3^2 + 4^2}} = \frac{|6 - 12 - 5|}{\sqrt{9 + 16}} = \frac{|-11|}{\sqrt{25}} = \frac{11}{5}$

Thus, the length of the latus rectum is $\frac{11}{5}$.

Step 2: Write the General Equation of the Family of Parabolas

We know the axis of the parabolas is parallel to the y-axis, and the length of the latus rectum is $\frac{11}{5}$.

Why this step? We need an equation representing all parabolas satisfying the given conditions.

The general equation is: $(x - h)^2 = 4a(y - k)$ Since $4a = \frac{11}{5}$, we have: $(x - h)^2 = \frac{11}{5}(y - k)$

Here, $h$ and $k$ are arbitrary constants representing the vertex coordinates.

Step 3: Differentiate the Equation Once with Respect to $x$

We differentiate both sides of $(x - h)^2 = \frac{11}{5}(y - k)$ with respect to $x$.

Why this step? Differentiating will help eliminate the constants $h$ and $k$.

Differentiating both sides: $\frac{d}{dx}(x - h)^2 = \frac{d}{dx}\left[\frac{11}{5}(y - k)\right]$ $2(x - h) = \frac{11}{5} \frac{dy}{dx}$

Step 4: Differentiate the Equation Again with Respect to $x$

We differentiate the equation $2(x - h) = \frac{11}{5} \frac{dy}{dx}$ with respect to $x$ again.

Why this step? We need to eliminate $h$, and another differentiation will achieve this.

Differentiating both sides: $\frac{d}{dx}[2(x - h)] = \frac{d}{dx}\left[\frac{11}{5} \frac{dy}{dx}\right]$ $2 = \frac{11}{5} \frac{d^2y}{dx^2}$

Step 5: Simplify the Equation

We simplify the equation $2 = \frac{11}{5} \frac{d^2y}{dx^2}$.

Why this step? To get the final form of the differential equation.

Multiplying both sides by $\frac{5}{11}$, we get: $\frac{d^2y}{dx^2} = \frac{10}{11}$ $11\frac{d^2y}{dx^2} = 10$

Common Mistakes & Tips

• Remember to use the chain rule when differentiating $y$ with respect to $x$.
• Ensure you have eliminated all arbitrary constants before arriving at the final differential equation.
• Double-check your differentiation and algebraic manipulations.

Summary

We started with the general equation of a parabola with its axis parallel to the y-axis, calculated the length of the latus rectum using the distance formula, and then differentiated the equation twice to eliminate the arbitrary constants $h$ and $k$. The resulting differential equation is $11\frac{d^2y}{dx^2} = 10$.

Final Answer

The final answer is \boxed{11{{{d^2}y} \over {d{x^2}}} = 10}, which corresponds to option (D).