Question

If the lines x = h and y = k is conjugate lines with respect to the hyperbola xy = c², then the locus of (h, k) is:

• A. xy = 2c²
• B. xy = c²
• C. xy = $\frac{c^{2}}{2}$
• D. None of these

Answer 1

Answer: (A)

xy = 2c²

Answer 2

The equation of polar is

xy₁ + yx₁= 2c²

Thus, the equation x – h = 0 and xy₁ + yx₁ = 2c² represent the same line

\ $\frac{y_{1}}{1}$= $\frac{x_{1}}{0}$= $\frac{2c^{2}}{h}$

Ž x₁ = 0, y₁ = $\frac{2c^{2}}{h}$

Since x = h and y = k are conjugate lines,

\ pole of the line x = h lies on y = k

\ y₁ – k = 0

Ž $\frac{2c^{2}}{h}$= k

Ž hk = 2c² or xy = 2c²