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1 016 MAΘ Natioal Covetio 1. Which Greek mathematicia do most historias credit with the discovery of coic sectios as a solutio to solvig the Delia problem, also kow as doublig the cube? Eratosthees Meaechmus Pythagoras Apolloius. How may poits i geeral liear positio are required to uiquely determie a coic sectio? What is the maximum umber of itersectios of two distict o-degeerate coic sectios? 3 6. Which of the followig cases is ot a possible degeerate coic sectio? Null set Poit Oe lie Two lies. Which coic sectio is described by the equatio 3x + xy + y -6x -6y +7= 0? Circle Ellipse Parabola Hyperbola 6. Let q be the acute agle of rotatio to stadard axes for the coic sectio give by the equatio 3x + xy + y -6x -6y +7= 0. Evaluate csc q. ( ) 1 + ( ) I order to rotate the coic sectio to a stadard axes through a agle q, which of the followig expressios is substituted for x? x'siq +y'cosq x'cosq -y'siq x'cosq +y'siq -x'siq +y'cosq 8. I the complex plae where z = x + yi,i = -1, ad z is the complex cojugate, which coic sectio is NOT described by the equatios iz iz, z z z 3, or z z z z? Circle Ellipse Parabola Hyperbola 9. As the eccetricity of a o-degeerate coic sectio approaches ifiity, what does the coic degeerate ito? Null set Poit 1 or lies poits
2 016 MAΘ Natioal Covetio 10. A coic sectio ca be writte as the matrix equatio, where ad é A B / D / ù ê ú A Q = ê B / C E / ú ê D / E / F ú ë û usig the coefficiets i geeral form. For the coic sectio described by the equatio 3x + xy + y -6x -6y +7= 0, let τ be the trace of AQ ad let Δ be the determiat. Evaluate τ -Δ The ext 0 questios are evely split four ways for each of the four coic sectios, deoted as either C (circle), E (ellipse), P (parabola), or H (hyperbola). 11. (C1) A circle has the equatio x + y +x - y += 0 with ceter (h,k). What is h+k? (E1) What is the area of the iscribed rectagle i a ellipse with equatio b x +a y = a b ad a>b>0 such that two of the sides of the rectagle are the latus recti, where c is the focal legth? ab c ab c a c b a c b 13. (P1) What is the miimum value of the parabola with equatio y = epx - x + 3 ep? e 1 ep ep 3 ep 1. (H1) If the eccetricity of a hyperbola is 3, what is the measure of the smaller agle betwee the two asymptotes, i degrees? ( What is the equatio of the circle iscribed i the triagle formed by the lies x=0, y=0, ad -x+3y=1? x + y +x -y +1= 0 x + y + 3x - 3y +3= 0 x + y + x - y + = 0 x + y +1x -1y +9 = 0
3 016 MAΘ Natioal Covetio 16. (E) What is the eccetricity of a ellipse i quadrat I of the Cartesia plae, taget to (,0) ad (0,3)? (P) What is the area of the figure eclosed by the parabola y -1x -6y -3= 0 ad its latus rectum?.p (H) A hyperbola is give by the equatio y -x -0x +8y -0 = 0. What is the product of the slope ad the x-itercept of the asymptote with egative slope? (C3) Lies are draw taget to the circle x + y =16 at the poits -, 3 What is the y-coordiate of the itersectio of the two taget lies? ( ) ad,- ( ). 0. (E3) A ellipse has foci located at (1,-) ad (,1) with eccetricity less tha 0.. Which of the followig poits caot exist o the ellipse? (1,) (-,1) (,) (-3,) 1. (P3) At what value of y does the lie taget to a parabola with equatio y +8x -10y +33= 0at the lower edpoit of its latus rectum itersect its directrix if said taget has slope 1? (H3) The Pell-Fermat equatio is a Diophatie equatio of the form x -y =1 for ay oegative iteger used to solve for iteger values of x ad y to approximate the square root of as x/y, so log that is ot a perfect square. It also happes to be a hyperbola i the Cartesia plae! What is the eccetricity of the hyperbola?
4 016 MAΘ Natioal Covetio 3. ( The circle with equatio x + y -3x +y +91= 0 is revolved about a lie i the Cartesia plae to geerate a torus with surface area p. If we cosider the solutio set of all lies that ca be the axis of rotatio to geerate such a torus, we geerate ifiitely may lies that are taget to a circle that is cocetric with the revolved circle. Thus, a aulus is formed from the two circles. What is the area eclosed by this aulus? p p 93p 36 8p. (E) Cosider a semi-elliptically-arched ceilig i a whisperig gallery. The vertical walls are of height feet, the ceilig reaches 0 feet above the vertical walls at its highest poit, ad the whisperig poits are located 30 feet across from each other at a height of feet. What is the height of the ceilig above the whisperig poits? (P) Cosider two distict poits o a arbitrary parabola P1 ad P with correspodig poits o the directrix Q1 ad Q such that P1Q1 ad PQ are perpedicular to the directrix, ad the focus of the parabola is F. How may of the followig statemets are always true? The distace from P1 to Q1 is the same as the distace from P to Q. The distace from P1 to Q1 is the same as the distace from P1 to F. The distace from P to Q is the same as the distace from Q to F. The distace from P1 to P is the same as the distace from Q1 to Q. The lie through P1 ad Q1 is parallel to the lie through P ad Q (H) A hyperbola has the equatio 9x -16y +18x +6y -199 = 0. What is the shortest distace from a focus of the hyperbola to either of its asymptotes? ( O the coordiate plae, a circle is formed from the three poits (,), (6,-) ad (,-). A equilateral triagle is the iscribed i the circle, ot ecessarily icludig ay of the above poits. What is the area eclosed by the equilateral triagle? (E) A ellipse has a focus (3,0), a directrix with equatio x+y-1=0, ad a eccetricity of 0.. Give the equatio of the ellipse i geeral quadratic form (positive x coefficiet, all coefficiets are relatively prime itegers), what is the costat term?
5 016 MAΘ Natioal Covetio 9. (P) A parabola has equatio y 3 6( x ). What is the sum of the x-itercepts of this parabola? (H) A hyperbola has polar equatio r =. What is the distace from a focus of this 1-3cosq hyperbola to the vertex of the parabola closer to this focus?
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