A tangent to the parabola `y^2 + 4bx = 0` meets the parabola `y^2 = 4ax` in P and Q. The locus of the middle points of PQ is:

Options

A. `x^2(2a+b)=4a^2y`
B. `y^2(2a+b)=4a^2x`
C. `y^2(a+b)=a^2x`
D. `x^2(a+b)=a^2y`

Answer

To find the locus of the midpoints of points P and Q where a tangent to the parabola \( y^2 + 4bx = 0 \) meets the parabola \( y^2 = 4ax \), we will follow these steps:

### Step 1: Understand the Parabolas
The first parabola \( y^2 + 4bx = 0 \) can be rewritten as \( y^2 = -4bx \), which opens to the left. The second parabola \( y^2 = 4ax \) opens to the right.

### Step 2: Equation of the Tangent
The equation of the tangent to the first parabola at a point can be expressed in slope form. For the parabola \( y^2 = -4bx \), the equation of the tangent line at a point \( (x_1, y_1) \) can be written as:
\[ y = mx - \frac{b}{m} \]
where \( m \) is the slope of the tangent.

### Step 3: Points of Intersection
We need to find the points of intersection \( P \) and \( Q \) of this tangent line with the second parabola \( y^2 = 4ax \). Substituting \( y = mx - \frac{b}{m} \) into \( y^2 = 4ax \) gives:
\[ (mx - \frac{b}{m})^2 = 4ax \]
Expanding this equation will lead to a quadratic in \( x \).

### Step 4: Midpoint of P and Q
Let the points of intersection be \( P(x_1, y_1) \) and \( Q(x_2, y_2) \). The midpoint \( M \) of \( PQ \) can be expressed as:
\[ M\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 5: Using the Properties of Quadratics
From the quadratic formed in Step 3, the sum of the roots \( x_1 + x_2 \) can be found using Vieta's formulas, which states that for a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots is given by \( -\frac{b}{a} \).

### Step 6: Locus of Midpoints
Using the relationships from the quadratic and the midpoint formula, we can express the coordinates of the midpoint \( M \) in terms of \( a \) and \( b \).

### Step 7: Final Locus Equation
After substituting and simplifying, we arrive at the locus equation:
\[ y^2(2a + b) = 4a^2x \]
This represents the locus of the midpoints of points \( P \) and \( Q \).

### Conclusion
The locus of the midpoints of \( P \) and \( Q \) is given by:
\[ y^2(2a + b) = 4a^2x \]