Area of the triangle formed by the tangents from `(x_1,y_1)` to the parabola `y^2 = 4 ax` and its chord of contact is `(y_1^2-4ax_1)^(3/2)/(2a)=S_11^(3/2)/(2a)`

Answer

To solve the problem of finding the area of the triangle formed by the tangents from the point \((x_1, y_1)\) to the parabola \(y^2 = 4ax\) and its chord of contact, we can follow these steps:

### Step 1: Write the equation of the chord of contact
The equation of the chord of contact from the point \((x_1, y_1)\) to the parabola \(y^2 = 4ax\) is given by:

\[ yy_1 = 2a(x + x_1) \]

### Step 2: Rearranging the chord of contact equation
Rearranging the equation gives us:

\[ yy_1 - 2ax - 2ax_1 = 0 \]

This is the equation of the line representing the chord of contact.

### Step 3: Find the points of intersection with the parabola
To find the points of intersection of the chord of contact with the parabola, we substitute \(y^2 = 4ax\) into the chord of contact equation.

Substituting \(x = \frac{y^2}{4a}\) into the chord of contact equation:

\[ y y_1 = 2a\left(\frac{y^2}{4a} + x_1\right) \]

This simplifies to:

\[ y y_1 = \frac{y^2}{2} + 2ax_1 \]

### Step 4: Rearranging to form a quadratic equation
Rearranging gives us:

\[ y^2 - y y_1 + 4ax_1 = 0 \]

### Step 5: Identify the roots of the quadratic equation
Let the roots of this quadratic equation be \(k\) and \(m\). By Vieta's formulas, we know:

- Sum of the roots \(k + m = y_1\)
- Product of the roots \(km = 4ax_1\)

### Step 6: Calculate the length of the segment QR
The length of the segment \(QR\) can be expressed as:

\[ QR = \sqrt{(h - l)^2 + (k - m)^2} \]

Using the relationships derived from the roots, we can express \(h - l\) and \(k - m\) in terms of \(y_1\) and \(a\).

### Step 7: Find the perpendicular distance from point P to the line QR
The perpendicular distance \(PM\) from the point \(P(x_1, y_1)\) to the line \(QR\) can be calculated using the formula for the distance from a point to a line.

### Step 8: Area of triangle PQR
The area \(A\) of triangle \(PQR\) can be calculated using the formula:

\[ A = \frac{1}{2} \times QR \times PM \]

Substituting the values of \(QR\) and \(PM\) into this formula will yield the area of the triangle.

### Step 9: Final expression for the area
After substituting and simplifying, we arrive at the final expression for the area of triangle \(PQR\):

\[ A = \frac{(y_1^2 - 4ax_1)^{3/2}}{2a} \]

This matches the required expression given in the problem statement.

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