The tangents to the parabola `y^2=4ax` at `P(at_1^2,2at_1)`, and `Q(at_2^2,2at_2)`, intersect at R. Prove that the area of the triangle PQR is `1/2a^2(t_1-t_2)^3`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the points P and Q
The points P and Q on the parabola \( y^2 = 4ax \) are given as:
- \( P(at_1^2, 2at_1) \)
- \( Q(at_2^2, 2at_2) \)

### Step 2: Write the equations of the tangents at points P and Q
The equation of the tangent to the parabola \( y^2 = 4ax \) at a point \( (at^2, 2at) \) is given by:
\[ ty = x + at^2 \]

Using this formula, we can write the equations of the tangents at points P and Q:
- Tangent at P:
\[ t_1y = x + at_1^2 \]
- Tangent at Q:
\[ t_2y = x + at_2^2 \]

### Step 3: Find the intersection point R of the tangents
To find the intersection point R, we need to solve the two equations simultaneously.

Rearranging the equations gives:
1. \( t_1y - x = at_1^2 \) (1)
2. \( t_2y - x = at_2^2 \) (2)

Subtracting the two equations (1) and (2):
\[ (t_1 - t_2)y = at_1^2 - at_2^2 \]
This simplifies to:
\[ (t_1 - t_2)y = a(t_1^2 - t_2^2) \]
Factoring the right side:
\[ (t_1 - t_2)y = a(t_1 - t_2)(t_1 + t_2) \]

Assuming \( t_1 \neq t_2 \), we can divide both sides by \( t_1 - t_2 \):
\[ y = a(t_1 + t_2) \]

Now substituting \( y \) back into one of the tangent equations to find \( x \):
Using the tangent at P:
\[ t_1(a(t_1 + t_2)) = x + at_1^2 \]
\[ at_1(t_1 + t_2) = x + at_1^2 \]
Rearranging gives:
\[ x = at_1(t_1 + t_2) - at_1^2 \]
\[ x = at_1t_2 \]

Thus, the coordinates of point R are:
\[ R(at_1t_2, a(t_1 + t_2)) \]

### Step 4: Calculate the area of triangle PQR
The area of triangle PQR can be calculated using the determinant formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Where:
- \( P(at_1^2, 2at_1) \)
- \( Q(at_2^2, 2at_2) \)
- \( R(at_1t_2, a(t_1 + t_2)) \)

Substituting these coordinates into the formula:
\[ \text{Area} = \frac{1}{2} \left| at_1^2(2at_2 - a(t_1 + t_2)) + at_2^2(a(t_1 + t_2) - 2at_1) + at_1t_2(2at_1 - 2at_2) \right| \]

### Step 5: Simplify the expression
Calculating each term:
1. First term:
\[ at_1^2(2at_2 - a(t_1 + t_2)) = at_1^2(2at_2 - at_1 - at_2) = at_1^2(a(t_2 - t_1)) \]
2. Second term:
\[ at_2^2(a(t_1 + t_2) - 2at_1) = at_2^2(a(t_2 - t_1)) \]
3. Third term:
\[ at_1t_2(2at_1 - 2at_2) = 2a(at_1t_2(t_1 - t_2)) \]

Combining these:
\[ \text{Area} = \frac{1}{2} \left| a(t_2 - t_1)(at_1^2 + at_2^2 - 2at_1t_2) \right| \]
Using the identity \( t_2^2 - t_1^2 = (t_2 - t_1)(t_2 + t_1) \):
\[ = \frac{1}{2} a^2 |(t_2 - t_1)(t_1 - t_2)(t_1 + t_2)| \]
\[ = \frac{1}{2} a^2 |(t_1 - t_2)^3| \]

Thus, we have shown that the area of triangle PQR is:
\[ \text{Area} = \frac{1}{2} a^2 (t_1 - t_2)^3 \]