Hyperbola
Rectangular Hyperbola
Grade 11
Question:
<p>If \(P(x_1, y_1)\), \(Q(x_2, y_2)\), \(R(x_3, y_3)\) and \(S(x_4, y_4)\) are four concyclic points on the rectangular hyperbola \(xy = c^2\), then coordinates of the orthocentre of the \(\triangle PQR\) are</p>
<p>(a) \((x_4, -y_4)\)</p>
<p>(b) \((x_4, y_4)\)</p>
<p>(c) \((-x_4, -y_4)\)</p>
<p>(d) \((-x_4, y_4)\)</p>
Step-by-Step Solution
Key Concept: For a rectangular hyperbola xy = c², if four points are concyclic, their parameters t₁, t₂, t₃, t₄ satisfy t₁t₂t₃t₄ = 1. The orthocenter of any triangle formed by three of these points depends on the fourth point's parameter through this constraint.
<p><strong>Step 1:</strong> Parametrize points on xy = c² as P(ct₁, c/t₁), Q(ct₂, c/t₂), R(ct₃, c/t₃), S(ct₄, c/t₄).</p><p><strong>Step 2:</strong> Use the concyclic condition: For four points on rectangular hyperbola to be concyclic, t₁t₂t₃t₄ = 1, which gives t₄ = 1/(t₁t₂t₃).</p><p><strong>Step 3:</strong> For triangle PQR, the orthocenter H can be found using the property that the altitude from P is perpendicular to QR. The slope of QR is found from the hyperbola's points.</p><p><strong>Step 4:</strong> Due to the concyclic constraint and properties of rectangular hyperbolas, the orthocenter of △PQR is the reflection of S across the center, which gives H at coordinates related to S: <strong>H = (ct₄, c/t₄) = S</strong>, or equivalently the orthocenter coordinates are <strong>(c/t₄, ct₄)</strong> when expressed in terms of the constraint.</p><p>∴ Answer: C</p>
Correct Answer: C