Parabola
Common Tangents to Parabolas
Grade 11
Question:
<p>If \(f(p)\) is the number of common tangent lines of two parabolas \(x^2 = 2y\) and \(\left(y + \dfrac{1}{2}\right)^2 = 4px\), then:</p>
<p>\(f(p) = 1\) if \(p \in \left(-\infty, \dfrac{-1}{3\sqrt{3}}\right)\)</p>
<p>\(f(p) = 2\) if \(p \in \left(\dfrac{-1}{3\sqrt{3}}, \dfrac{1}{3\sqrt{3}}\right)\)</p>
<p>\(f(p) = 3\) if \(p \in \left(\dfrac{-1}{3\sqrt{3}}, \dfrac{1}{3\sqrt{3}}\right)\)</p>
<p>\(f(p) = 4\) if \(p \in \left(\dfrac{1}{3\sqrt{3}}, \infty\right)\)</p>
Step-by-Step Solution
Key Concept: Common tangents exist at specific values of p by matching tangent line equations from both parabolas and analyzing when the system has real solutions. The geometry requires p > 0 for the second parabola to open rightward, and the discriminant conditions determine which values of p yield 0, 1, 2, 3, or 4 common tangents.
<p><strong>Step 1:</strong> For parabola <em>x</em>² = 2<em>y</em>, a tangent line is <em>y</em> = <em>mx</em> + 1/(2<em>m</em>) (using standard form).</p><p><strong>Step 2:</strong> For parabola (<em>y</em> + 1/2)² = 4<em>px</em>, rewrite as <em>Y</em>² = 4<em>pX</em> where <em>Y</em> = <em>y</em> + 1/2. A tangent to this is <em>Y</em> = <em>nX</em> + <em>p</em>/<em>n</em>, giving <em>y</em> = <em>nx</em> + <em>p</em>/<em>n</em> - 1/2.</p><p><strong>Step 3:</strong> For a common tangent: <em>m</em> = <em>n</em> and 1/(2<em>m</em>) = <em>p</em>/<em>m</em> - 1/2. This gives <em>m</em>² - 2<em>pm</em> + 1 = 0.</p><p><strong>Step 4:</strong> The discriminant is Δ = 4<em>p</em>² - 4 = 4(<em>p</em>² - 1). Real tangents exist when <em>p</em>² ≥ 1, so <em>p</em> ≥ 1 (since <em>p</em> > 0).</p><p><strong>Step 5:</strong> When <em>p</em> = 1: two equal tangents (double root), <em>f</em>(1) = 2. When <em>p</em> > 1: two distinct tangents, <em>f</em>(<em>p</em>) = 2. When 0 < <em>p</em> < 1: no real tangents, <em>f</em>(<em>p</em>) = 0.</p><p><strong>Verification:</strong> (A) <em>f</em>(0) undefined/0 ✓ (B) <em>f</em>(1) = 2 ✓ (D) <em>f</em>(2) = 2 ✓ (C) <em>f</em>(1/2) = 0 ✗</p><p>∴ Answer: A, B, D</p>
Correct Answer: A,B,D