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Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Let (-a,0) be the left vertex of the ellipse. What is the distance from this vertex to F1?
  1. √(c-a)2+y2
  2. √a2+c2
  3. a
  4. a-c
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Which of the following is the best definition of an ellipse?
  1. A squished circle.
  2. An oblong shaped closed curve that has two foci.
  3. Given two points, the foci, the set of all points where the sum of the the distances between a given point and the foci is constant.
  4. The set of all points that are bounded by the distances from two central points, foci, and the major and minor axes.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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The method used in the previous questions is the only way to derive the equation of a circle.
  1. True
  2. False
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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In the first question, a right triangle was created, using the radius of the circle as the hypotenuse, such that one leg had a slope of zero, and the other leg's slope was undefined. Why was this?
  1. This is the only right triangle that can be formed for the given hypotenuse.
  2. This is the only right triangle that can be formed for the given hypotenuse that is completely inside the given circle.
  3. Using a right triangle with legs that have either a slope of zero or an undefined slope means that they represent the vertical and horizontal distance of the hypotenuse.
  4. There is no need for this triangle, but it makes the question more difficult because it results in two possible correct answers.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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In the first question, the stipulation on the point (x,y) was made that the x-coordinate could not be 3, and the y-coordinate could not be 4. Why was this?
  1. These points do not actually occur on the circle, and so would have caused an error in the derivation.
  2. If a point met either of these criteria, it would be either directly above, below, to the right of, or to the left of the center of the circle. This would have made creating a right triangle with legs that have slope of 0 and undefined impossible.
  3. These points would have resulted in complex values (calculations involving imaginary numbers), and led to serious complications in the derivation process.
  4. The points that meet these criteria result in triangles which are not right triangles.
Grade 11 Nonlinear Equations and Functions
Identify the equation.
x2-6xy+y2-10x-5y+11=0
  1. Hyperbola
  2. Circle
  3. Parabola
  4. Ellipse
Grade 11 Nonlinear Equations and Functions
Identify the center and the radius.
(x-6)2+(y-4)2=9.
  1. Center (6, 4), radius 9
  2. Center (-6, -4), radius 3
  3. Center (6, 4), radius 3
Grade 12 Nonlinear Equations and Functions CCSS: HSA-REI.A.2
p+5p2+p=1p2+p-p-6p+1
  1. -4
  2. 6,8
  3. 5,5
  4. 4,1
Grade 11 Nonlinear Equations and Functions CCSS: HSA-REI.A.2
Solve for all possible values of x. 6x-12x-2=6-x
  1. 0
  2. 2, 12
  3. 2, 0
  4. No Solution
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Which of the following is the best definition of a hyperbola?
  1. Two opposing parabolas.
  2. Given two points, called foci, the set of points such that the absolute value of the differences of the distance between the foci and a point is constant.
  3. The distances between a set of points and the foci is constant.
  4. Two points, foci, whose eccentricity is greater than 1.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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After squaring both sides, expanding, combing like terms, and dividing by any common constants, what is the resulting equation?
  1. xc-a2=-a√(x-c)2+y2
  2. 2a2+c2+x2+y2=-2a√(x-c)2+y2
  3. -2a2+c2+x2+y2=0
  4. a2=cx
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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What is the distance between (-a,0) and F2?
  1. a+c
  2. c-a
  3. a-c
  4. √a2-c2
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Let (x,y) be a point on the ellipse. Which equation represents the distance, d1, between F1 and the point (x,y)?
  1. d1=√(x+c)2+y2
  2. d1=√(x-c)+y2
  3. d1=√(-x+c)2+y2
  4. d1=√(x-c)2+(y-x)2
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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For the points which are excluded by the condition mentioned in the previous question, does the equation of the circle not work for them? Why or why not?
  1. It doesn't, because either the x or y distance becomes zero meaning an entirely different equation is required.
  2. It doesn't, these points are always excluded, but just usually ignored.
  3. It does, however these points would just have introduced complex numbers into the derivation and so were avoided.
  4. It does, these points are simply special cases where the distance from the center to the circle is all in either the x or y direction.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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Now that we have the lengths of the legs of the triangle, what theorem, property, or formula correctly relates them and why is it used?
  1. The formula for the area of a triangle, A=12(y-3)⋅(x-4), since we know the height and base of the triangle.
  2. Heron's Formula, A=√s⋅(s-a)⋅(s-b)⋅(s-c), where s=2+(y-4)2+(x-3)22, because the length of all three sides is known.
  3. Pythagorean Theorem, (x-3)2+(y-4)2=4, since the lengths of the three sides of the right triangle are known, and this introduces no extra variables.
  4. Law of Cosines, |x-3|2=4+|y-4|2-2⋅2⋅|y-4|⋅cos90°, because the lengths of the three sides of the triangle, and the measure of the right angle, are known, and this introduces no extra variables.
Grade 11 Nonlinear Equations and Functions
4x2+4y2-8x+12y-32=7 is what type of conic?
  1. Parabola
  2. Ellipse
  3. Circle
  4. Hyperbola
Grade 12 Nonlinear Equations and Functions CCSS: HSA-REI.A.2
16k2=13k2-1k
  1. k
  2. 1/6
  3. -2
  4. 21
Grade 11 Nonlinear Equations and Functions
What is the center of the following hyperbola, ((x-3)225-(y+5)24)=1?
  1. (-3,5)
  2. (3,-5)
  3. (-5,3)
  4. (5,-3)
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