Question

If P (x₁, y₁), Q(x₂, y₂), R(x₃, y₃) and S(x₄, y₄) are four concyclic points on the rectangular hyperbola xy = c², the coordinates of orthocentre of the ∆PQR are:

• A. (x₄, -y₄)
• B. (x₄, y₄)
• C. (-x₄, -y₄)
• D. (-x₄, y₄)

Answer 1

Answer: (A)

(x₄, -y₄)

Answer 2

Since, points (x_r, y_r) is lying on xy = c² for r = 1, 2, 3, 4

∴ $y_{r} = \frac{c^{2}}{x_{r}}$ for r = 1, 2, 3, 4

Slope of QR is - $\frac{c^{2}}{x_{2}x_{3}}$.

∴ Equation of line passing through A and perpendicular to QR is

$x_{1}x_{2}x_{3}x - c^{2}x_{1}y = {x_{1}}^{2}x_{2}x_{3} - c^{4}$... (1)

Similarly, equation of line through Q and perpendicular to PR is

$x_{1}x_{2}x_{3}x - c^{2}x_{2}y = x_{1}{x_{2}}^{2}x_{3} - c^{4}$ ... (2)

(1) - (2) ⇒ y = $- \frac{x_{1}x_{2}x_{3}}{c^{2}}$

∴ $x = \frac{c^{4}}{x_{1}x_{2}x_{3}}$.

Thus, orthocentre is again lying on xy = c² i.e. the fourth point (x₄, y₄).