6.4 Q-5

Question Statement

Prove that the ordinate ($y$) at any point $P(x, y)$ on a parabola is the mean proportional between the length of the latus rectum and the abscissa ($x$) of $P$.

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Background and Explanation

To solve this problem, we need to understand:

1. The standard equation of a parabola:

$$y^2 = 4ax$$

where $a$ is the distance from the vertex to the focus.

2. The latus rectum of a parabola:

   • It is the line segment perpendicular to the axis of symmetry passing through the focus.
   • Its length is $4a$.

3. The mean proportional: If a number $b$ is the mean proportional between two numbers $a$ and $c$, then:

$$b^2 = ac \quad \Rightarrow \quad b = \sqrt{ac}$$

The goal is to show that for any point $P(x, y)$ on the parabola $y^2 = 4ax$, the ordinate $y$ satisfies this property.

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Solution

Step 1: Write the Standard Equation of the Parabola

The standard form of the parabola is:

$$y^2 = 4ax$$

Here:

• $y$ is the ordinate of the point $P(x, y)$.
• $x$ is the abscissa of $P(x, y)$.

Step 2: Express the Ordinate ($y$) in Terms of $x$

From the equation of the parabola:

$$y^2 = 4ax$$

Taking the square root on both sides:

$$y = \pm \sqrt{4ax}$$

Step 3: Interpret the Result

The expression for $y$ can be rewritten as:

$$y = \pm \sqrt{(4a) \cdot x}$$

Here:

• $4a$ is the length of the latus rectum.
• $x$ is the abscissa of the point $P(x, y)$.

This shows that $y$ is the mean proportional between the length of the latus rectum ($4a$) and the abscissa ($x$) of $P(x, y)$.

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Key Formulas or Methods Used

1. Equation of the Parabola:

$$y^2 = 4ax$$

2. Mean Proportional: If $b$ is the mean proportional between $a$ and $c$, then:

$$b = \sqrt{ac}$$

3. Length of the Latus Rectum: For a parabola $y^2 = 4ax$, the latus rectum length is:

$$4a$$

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Summary of Steps

1. Start with the standard form of the parabola: $y^2 = 4ax$.
2. Solve for $y$ to get: $y = \pm \sqrt{4ax}$.
3. Recognize that $y$ is expressed as the square root of the product of $4a$ (latus rectum) and $x$ (abscissa).
4. Conclude that the ordinate $y$ is the mean proportional between the latus rectum length and the abscissa $x$.