Question

The tangent and normal to a rectangular hyperbola $xy=c²$ at a point cut off intercepts $a₁, a₂$ on one axis and $b₁, b₂$ on the other, then $a₁a₂ + b₁b₂$ is equal to

• A. 1
• B. 2
• C. 3
• D. 0

Hint:

The tangent and normal to a rectangular hyperbola $xy=c$ at a point cut off intercepts $a1, a2$ on one axis and $b1, b2$ on the other, then $a1a2 + b1b2$ is equal to

Answer 1

The Correct Option is D

Step 1: Concept
Equation of tangent at $(ct, c/t)$ is $x + yt^2 = 2ct$. Intercepts are $a_1 = 2ct, b_1 = 2c/t$.
Step 2: Analysis
Equation of normal is $xt^3 - yt = ct^4 - c$. Intercepts are $a_2 = c(t - 1/t^3), b_2 = c(1/t - t^3)$.
Step 3: Evaluation
Multiply intercepts: $a_1a_2 = 2c^2t(t - 1/t^3) = 2c^2(t^2 - 1/t^2)$.
$b_1b_2 = \frac{2c}{t} \cdot c(1/t - t^3) = 2c^2(1/t^2 - t^2)$.
Step 4: Conclusion
$a_1a_2 + b_1b_2 = 2c^2(t^2 - 1/t^2 + 1/t^2 - t^2) = 0$.
Final Answer: (d)