What is the locus of the vertices of all the quadratic equations whose roots are m times the certain given roots say `alpha` and `beta`?

Options

A. Polygon
B. Parabola
C. A line parallel to X-axis
D. Nothing can be said

Answer

To find the locus of the vertices of all quadratic equations whose roots are \( m \) times the given roots \( \alpha \) and \( \beta \), we can follow these steps:

### Step 1: Understand the roots of the quadratic equation
The roots of the quadratic equation can be expressed as:
\[ r_1 = m\alpha \quad \text{and} \quad r_2 = m\beta \]

### Step 2: Write the quadratic equation
Using the roots, we can write the quadratic equation in the standard form:
\[ x^2 - (r_1 + r_2)x + r_1 r_2 = 0 \]
Substituting the values of \( r_1 \) and \( r_2 \):
\[ x^2 - (m\alpha + m\beta)x + (m\alpha)(m\beta) = 0 \]
This simplifies to:
\[ x^2 - m(\alpha + \beta)x + m^2 \alpha \beta = 0 \]

### Step 3: Identify the vertex of the quadratic equation
The vertex \( V \) of a quadratic equation in the form \( ax^2 + bx + c \) is given by the formula:
\[ V = \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) \]
For our equation:
- \( a = 1 \)
- \( b = -m(\alpha + \beta) \)

Calculating the x-coordinate of the vertex:
\[ x_V = -\frac{-m(\alpha + \beta)}{2 \cdot 1} = \frac{m(\alpha + \beta)}{2} \]

### Step 4: Calculate the y-coordinate of the vertex
To find the y-coordinate, substitute \( x_V \) back into the quadratic equation:
\[ f\left(\frac{m(\alpha + \beta)}{2}\right) = \left(\frac{m(\alpha + \beta)}{2}\right)^2 - m(\alpha + \beta)\left(\frac{m(\alpha + \beta)}{2}\right) + m^2\alpha\beta \]
This simplifies to:
\[ f\left(\frac{m(\alpha + \beta)}{2}\right) = \frac{m^2(\alpha + \beta)^2}{4} - \frac{m^2(\alpha + \beta)^2}{2} + m^2\alpha\beta \]
Combining the terms gives:
\[ f\left(\frac{m(\alpha + \beta)}{2}\right) = m^2\alpha\beta - \frac{m^2(\alpha + \beta)^2}{4} \]

### Step 5: Determine the locus
The vertex \( V \) can thus be expressed as:
\[ V = \left( \frac{m(\alpha + \beta)}{2}, m^2\alpha\beta - \frac{m^2(\alpha + \beta)^2}{4} \right) \]
To find the locus, we can express this in terms of coordinates \( (h, k) \):
- Let \( h = \frac{m(\alpha + \beta)}{2} \)
- Let \( k = m^2\alpha\beta - \frac{m^2(\alpha + \beta)^2}{4} \)

This gives us a relationship between \( h \) and \( k \) that defines the locus of the vertices.

### Final Result
The locus of the vertices of all the quadratic equations whose roots are \( m \) times the given roots \( \alpha \) and \( \beta \) can be expressed as a curve in the \( (h, k) \) plane.

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