Zadanie 2d, Paper 3, listopad 2023, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2023, Paper 3 (zadania wzorowane), pytanie 2(d). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
This investigation concerns intersecting curve families. In 2a, take parabolas y=kx², where k≠0; work locally where x,y≠0. In 2b–2d use Ea : x2/(a2 + 9) + y2/a2 = 1 for a>0 and Hb : x2/(9 − b2) − y2/b2 = 1 for 0<b<3. In 2e–2f a tangent of finite slope f is rotated anticlockwise through an angle α; where its new slope is finite, .
Polecenie
Show that E_a and H_b intersect orthogonally at their first-quadrant intersection M. You may use xM2 = (a2 + 9)(9 − b2)/9 and yM2 = a2b2/9.
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Odpowiedź z klucza
Implicit differentiation gives slopes mE = −a2x/[(a2 + 9)y] and mH = b2x/[(9 − b2)y]. At M both x,y are nonzero, so mEmH = −a2b2xM2/[(a2 + 9)(9 − b2)yM2] = −1 after using the given squared coordinates. Thus the tangents, and hence the curves, meet at right angles.
Schemat punktowania
M1:implicitly differentiate either curve (or differentiate an explicit local branch). A1:both correct slope expressions. M1:substitute the supplied intersection coordinates or squared-coordinate ratio. A1:correct substituted product/slope values. A1:product −1 and the orthogonality conclusion. The final A1 is earned by a correct product equal to −1, an explicit negative-reciprocal relationship, or the equivalent statement that the two obtained slopes have product −1. The concluding orthogonality sentence is the supplied result and carries no additional requirement.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Exact claims and proofs require analytical working. Counts of intercepts mean distinct real points, unless repeated roots are explicitly requested.