Question

If m₁ and m₂ are the slopes of the tangents to the hyperbola 16x² –25y² = 400 which pass through the point (6, 2) then the harmonic mean of m₁ and m₂ is-

• A. $\frac{3}{5}$
• B. $\frac{- 3}{5}$
• C. $\frac{- 5}{3}$
• D. $\frac{5}{3}$

Answer 1

Answer: (D)

$\frac{5}{3}$

Answer 2

16x² – 25y² = 400 Ž $\frac{x^{2}}{25}$ – $\frac{y^{2}}{16}$= 1

equation of tangent Ž y = mx ± $\sqrt{25m^{2}–16}$

it passes through the point (6, 2)

2 = 6m ± $\sqrt{25m^{2}–16}$Ž (2 – 6m)² = 25m² – 16

4 + 36m² – 24m = 25m² – 16

Ž 11m² – 24m +20 = 0

m₁ + m₂ = 24/11; m₁ m₂ = 20/11

Harmonic mean of m₁ & m₂

H = $\frac{2m_{1}m_{2}}{m_{1} + m_{2}}$ = $\frac{2 \times \frac{20}{11}}{24/11}$Ž H = $\frac{5}{3}$