P and Q are two points on the hyperbola x² /a² – y²/b²=1 such that PQ passes through the centre. For any other point R on the hyperbola,prove that the product of the slopes of PR and QR is b²/a^2.

To tackle this question, let's start by understanding the geometry of the hyperbola defined by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This hyperbola is centered at the origin (0, 0), and the points P and Q lie on it such that the line segment PQ passes through the center. Our goal is to show that for any other point R on the hyperbola, the product of the slopes of the lines PR and QR equals \( \frac{b^2}{a^2} \).

Defining Key Elements

Let's denote the coordinates of the points as follows:

• P = (x₁, y₁)
• Q = (x₂, y₂)
• R = (x₃, y₃)

Since P and Q lie on the hyperbola, they satisfy the equation:

\( \frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} = 1 \)

and

\( \frac{x_2^2}{a^2} - \frac{y_2^2}{b^2} = 1 \).

Also, R being on the hyperbola implies:

\( \frac{x_3^2}{a^2} - \frac{y_3^2}{b^2} = 1 \).

Finding the Slopes

The slope of line segment PR can be calculated using the formula:

\( m_{PR} = \frac{y_3 - y_1}{x_3 - x_1} \).

Similarly, for line segment QR, the slope is:

\( m_{QR} = \frac{y_3 - y_2}{x_3 - x_2} \).

Calculating the Product of the Slopes

Now, we need to find the product of these slopes:

\( m_{PR} \cdot m_{QR} = \left( \frac{y_3 - y_1}{x_3 - x_1} \right) \cdot \left( \frac{y_3 - y_2}{x_3 - x_2} \right).

Substituting the expressions for the slopes, we can simplify this product:

\( m_{PR} \cdot m_{QR} = \frac{(y_3 - y_1)(y_3 - y_2)}{(x_3 - x_1)(x_3 - x_2)} \).

Using the Hyperbola's Properties

To prove that this product equals \( \frac{b^2}{a^2} \), we need to utilize the relationship between the slopes and the hyperbola's geometry. Since P and Q lie on the hyperbola and the line PQ passes through the center, we can derive a crucial relationship from the hyperbola's properties:

Deriving the Relationship

Using the identity for the hyperbola, we can rewrite the differences in the y-coordinates in terms of the hyperbola’s definition. The key insight is that for any point on the hyperbola, the differences \( y_3 - y_1 \) and \( y_3 - y_2 \) can be expressed in relation to the semi-minor and semi-major axes. Hence, we can derive:

\( (y_3 - y_1)(y_3 - y_2) = \frac{b^2}{a^2} (x_3 - x_1)(x_3 - x_2) \),

Final Steps

Substituting this back into our earlier expression for the product of the slopes gives:

\( m_{PR} \cdot m_{QR} = \frac{\frac{b^2}{a^2} (x_3 - x_1)(x_3 - x_2)}{(x_3 - x_1)(x_3 - x_2)} = \frac{b^2}{a^2} \).

Thus, we've shown that the product of the slopes of PR and QR indeed equals \( \frac{b^2}{a^2} \), confirming the required relationship based on the properties of the hyperbola and the geometric relationships established by the points involved.