<p>If a normal to a rectangular hyperbola at a point cuts off intercepts \(a_1, a_2\) on one axis and \(b_1, b_2\) on the other axis then \(a_1 a_2 + b_1 b_2 =\) ___.</p>
Step-by-Step Solution
Key Concept: For a rectangular hyperbola xy = c², the normal at any point has a special property: when it intersects the coordinate axes at four points forming intercepts a₁, a₂ on one axis and b₁, b₂ on the other, the sum of products of these intercepts equals a constant related to the hyperbola's parameter.
<p><strong>Step 1:</strong> For rectangular hyperbola xy = c², use parametric form: point P = (ct, c/t)</p><p><strong>Step 2:</strong> The equation of normal at P(ct, c/t) is: t³x - ty + c(t⁴ - 1) = 0</p><p><strong>Step 3:</strong> Find x-axis intercepts (y = 0): x = c(1 - t⁴)/t³, giving two intercepts a₁, a₂</p><p><strong>Step 4:</strong> Find y-axis intercepts (x = 0): y = c(t⁴ - 1)/t, giving two intercepts b₁, b₂</p><p><strong>Step 5:</strong> Calculate a₁a₂ from the normal equation coefficients. The product of intercepts on x-axis: a₁a₂ = -c²</p><p><strong>Step 6:</strong> Calculate b₁b₂ from the normal equation coefficients. The product of intercepts on y-axis: b₁b₂ = c²</p><p><strong>Step 7:</strong> Therefore a₁a₂ + b₁b₂ = -c² + c² = 0</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0