If `P(at_1^2,2at_1)`and `Q(at_2^2,2at_2)` are two variablé points on the curve `y^2 = 4ax` and PQ subtends a right angle at the vertex, then `t_1t_2` , is equal to

Options

A. `-1`
B. `-2`
C. `-3`
D. `-4`

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Identify the points on the parabola
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: Determine the coordinates of the vertex
The vertex of the parabola \( y^2 = 4ax \) is at the origin, which is \( O(0, 0) \).

### Step 3: Calculate the slopes of lines OP and OQ
The slope of line \( OP \) from the origin \( O \) to point \( P \) is given by:
\[ \text{slope of } OP = \frac{y_P - y_O}{x_P - x_O} = \frac{2at_1 - 0}{at_1^2 - 0} = \frac{2a t_1}{a t_1^2} = \frac{2}{t_1} \]

Similarly, the slope of line \( OQ \) from the origin \( O \) to point \( Q \) is:
\[ \text{slope of } OQ = \frac{y_Q - y_O}{x_Q - x_O} = \frac{2at_2 - 0}{at_2^2 - 0} = \frac{2a t_2}{a t_2^2} = \frac{2}{t_2} \]

### Step 4: Use the condition for perpendicular lines
Since \( PQ \) subtends a right angle at the vertex \( O \), the product of the slopes of lines \( OP \) and \( OQ \) must equal \(-1\):
\[ \left(\frac{2}{t_1}\right) \left(\frac{2}{t_2}\right) = -1 \]

### Step 5: Simplify the equation
This simplifies to:
\[ \frac{4}{t_1 t_2} = -1 \]

### Step 6: Solve for \( t_1 t_2 \)
Multiplying both sides by \( t_1 t_2 \) gives:
\[ 4 = -t_1 t_2 \]
Thus,
\[ t_1 t_2 = -4 \]

### Final Answer
The value of \( t_1 t_2 \) is \( -4 \).

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