Question

The line 3x + 2y + 1 = 0 meets the hyperbola 4x² - y² = 4a² in the points P and Q. The coordinates of the point of intersection of the tangents at P and Q are+

• A. (-3a², 8a²)
• B. (3a², 8a²)
• C. (3a², -8a²)
• D. None of these

Answer 1

Answer: (A)

(-3a², 8a²)

Answer 2

The required point is clearly the pole of the given line.

Let (x₁, y₁) be the pole of 3x + 2y + 1 = 0 ... (1)

w.r.t. the hyperbola 4x² - y² = 4a² ... (2)

Polar of (x₁, y₁) w.r.t. (2) is 4xx₁ - yy₁ = 4a² ... (3)

As (1) and (3) represent the same line i.e. polar of (x₁, y₁),

∴ comparing coefficients, we get

$\frac{4x_{1}}{3} = \frac{- y_{1}}{2} = \frac{4a^{2}}{- 1} \Rightarrow x_{1} - 3a^{2},y_{1} = 8a^{2}$.

Hence the required point i.e. the pole of (1) is (-3a², 8a²).