Tangents are drawn from points of the parabola `y^(2)=4ax` to the parabola `y^(2)=4b(x-c)`. Find the locus of the mid point of chord of contact.

Answer

To solve the problem of finding the locus of the midpoint of the chord of contact of tangents drawn from points on the parabola \(y^2 = 4ax\) to the parabola \(y^2 = 4b(x - c)\), we will follow these steps:

### Step 1: Understand the Parabolas
The first parabola is given by \(y^2 = 4ax\). This is a standard parabola that opens to the right. The second parabola is given by \(y^2 = 4b(x - c)\), which is also a right-opening parabola but is shifted to the right by \(c\).

### Step 2: Find the Equation of the Tangent
For a point \((x_1, y_1)\) on the first parabola \(y^2 = 4ax\), the equation of the tangent to the parabola is given by:
\[ yy_1 = 2a(x + x_1) \]

### Step 3: Substitute into the Second Parabola
For the second parabola \(y^2 = 4b(x - c)\), we need to express the condition for the tangents drawn from \((x_1, y_1)\) to touch this parabola. The chord of contact can be expressed as:
\[ yy_1 = 2b(x - c) \]

### Step 4: Set Up the Locus of the Midpoint
Let the midpoint of the chord of contact be \((h, k)\). The equations of the tangents can be rewritten in terms of \(h\) and \(k\):
\[ k y_1 = 2a(h + x_1) \quad \text{and} \quad k y_1 = 2b(h - c) \]

### Step 5: Relate the Two Equations
Since both expressions equal \(k y_1\), we can set them equal to each other:
\[ 2a(h + x_1) = 2b(h - c) \]

### Step 6: Solve for \(x_1\)
Rearranging gives:
\[ ah + ax_1 = bh - bc \]
\[ ax_1 = bh - ah - bc \]
\[ x_1 = \frac{bh - ah - bc}{a} \]

### Step 7: Substitute Back into the Parabola Equation
Now substitute \(x_1\) back into the first parabola equation \(y_1^2 = 4ax_1\):
\[ y_1^2 = 4a\left(\frac{bh - ah - bc}{a}\right) \]
This simplifies to:
\[ y_1^2 = 4(bh - ah - bc) \]

### Step 8: Express \(y_1\) in Terms of \(h\) and \(k\)
From the chord of contact, we also have:
\[ k^2 = 4b(h - c) \]

### Step 9: Equate and Simplify
Now we have two expressions for \(y_1^2\):
1. \(y_1^2 = 4(bh - ah - bc)\)
2. \(k^2 = 4b(h - c)\)

Setting these equal gives us the locus equation:
\[ 4(bh - ah - bc) = 4b(h - c) \]

### Step 10: Final Locus Equation
After simplifying, we find:
\[ b(h - c) = ah \]
This leads to the final locus equation:
\[ bh - ah = bc \quad \Rightarrow \quad (b - a)h = bc \]
Thus, the locus of the midpoint of the chord of contact is:
\[ h = \frac{bc}{b - a} \]

### Summary of Steps
1. Understand the equations of the parabolas.
2. Write the equation of the tangent from a point on the first parabola.
3. Set up the condition for tangents to the second parabola.
4. Express the midpoint of the chord of contact.
5. Relate the two equations and solve for \(x_1\).
6. Substitute back into the parabola equation.
7. Equate and simplify to find the locus.