Ready to dive into the ultimate conics test? Whether you're mastering math for class or prepping for exams, our conic sections quiz will challenge your understanding of ellipses, parabolas and hyperbolas. In this quiz on conic sections you'll not only check your grasp of each curve's key properties, but also gain confidence tackling real problems. This free, interactive conic curves challenge is perfect for students, teachers or anyone curious about these elegant shapes. Hone your skills with engaging conic sections practice problems and sharpen your focus with a fun hyperbolas quiz part 1 . Take the leap, start your practice conic test now and prove you're a conics pro!
Which conic section is defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix)?
Ellipse
Hyperbola
Circle
Parabola
A parabola is precisely the locus of points equidistant from a focus and a directrix line. This definition distinguishes it from other conics like ellipses or hyperbolas. See more details at MathWorld.
What is the standard equation of a circle centered at the origin with radius r?
x^2 + y^2 = r^2
x^2 - y^2 = r^2
(x - r)^2 + (y - r)^2 = r^2
x^2 + y^2 = r
A circle of radius r centered at the origin satisfies x^2 + y^2 = r^2 because each point on the circle is exactly r units from (0,0). This is the standard form found in most precalculus texts. For more, see Khan Academy.
Which conic section has an eccentricity equal to zero?
Parabola
Circle
Hyperbola
Ellipse
The eccentricity e measures how stretched a conic is. A circle has e = 0 because it is perfectly round, with no elongation. Learn more at MathWorld.
What is the range of eccentricity for an ellipse?
e = 1
e > 1
0 < e < 1
-1 < e < 0
An ellipse has eccentricity between 0 and 1, where 0 corresponds to a circle and values approaching 1 become more elongated. This is a standard property of ellipses. More at Britannica.
What is the standard form of a parabola opening to the right with vertex at the origin?
x^2 = 4py
y^2 = px
x = 4py
y^2 = 4px
A right?opening parabola with vertex at (0,0) has the form y^2 = 4px, where p is the focal length. This places the focus at (p,0). See MathWorld for details.
Which conic section consists of two disconnected curves called branches?
Parabola
Circle
Hyperbola
Ellipse
A hyperbola is made up of two separate branches, each approaching its asymptotes at infinity. This distinct shape differentiates it from ellipses and parabolas. More at Khan Academy.
The sum of the distances from any point on the curve to the two foci is constant. Which conic does this describe?
Circle
Parabola
Hyperbola
Ellipse
An ellipse is defined by the property that the sum of distances from any point on it to two fixed foci is constant. This distinguishes it from hyperbolas, which use a difference. See Wikipedia.
Given the equation (x-2)^2/9 + (y+3)^2/16 = 1, what is the orientation of the major axis?
Neither
Horizontal
Vertical
At 45 degrees
Because the larger denominator (16) is under the (y+3)^2 term, the major axis is vertical. In standard ellipses, the major axis aligns with the variable having the larger denominator. For more, see MathBitsNotebook.
What is the eccentricity of the ellipse defined by 4x^2 + 9y^2 = 36?
3/5
2/3
?13/6
?5/3
Rewrite as x^2/9 + y^2/4 = 1 so a=3, b=2. Then e = ?(1 - b^2/a^2) = ?(1 - 4/9) = ?5/3. More at Khan Academy.
For the parabola y^2 = 12x, what is the equation of its directrix?
y = -3
x = -3
x = 3
y = 3
In y^2 = 4px, 4p = 12 so p = 3, giving a directrix x = -p = -3. See MathWorld.
What is the length of the latus rectum of the parabola y^2 = 8x?
4
16
8
2
The latus rectum of y^2 = 4px has length 4p. Here 4p = 8, so p = 2, and the latus rectum = 4p = 8. More at Math is Fun.
Find the distance between the foci of the hyperbola x^2/16 - y^2/9 = 1.
6
12
10
8
For hyperbola x^2/a^2 - y^2/b^2 =1, c^2 = a^2 + b^2 = 16 + 9 =25, c =5, so the total distance between foci = 2c =10. See Wikipedia.
What are the coordinates of the focus of the parabola y = (1/8)x^2?
(0, 2)
(2, 0)
(0, -2)
(-2, 0)
Rewriting y = x^2/(4p) gives 4p = 8 so p = 2, placing the focus at (0, p) = (0,2). See Khan Academy.
What is the eccentricity of a hyperbola?
e = 0
0 < e < 1
e > 1
e < 0
Hyperbolas always have eccentricity greater than 1, distinguishing them from ellipses and circles. More at MathWorld.
Determine the asymptotes of the hyperbola y^2 - x^2 = 1.
y = 2x and y = -2x
y = x^2
y = x and y = -x
y = -x^2
Rewrite as y^2/x^2 - 1 = 1/x^2; for large values, behavior approaches y^2/x^2 = 1, so y = x. See Lamar University Tutorial.
What is the equation of the ellipse with foci at (-4,0) and (4,0) and a major axis length of 10?
x^2/9 + y^2/25 = 1
x^2/16 + y^2/9 = 1
x^2/25 - y^2/9 = 1
x^2/25 + y^2/9 = 1
Here a = 5 (half of 10), c = 4, so b^2 = a^2 - c^2 = 25 - 16 = 9. The standard form is x^2/25 + y^2/9 = 1. See Math is Fun.
Compute the discriminant B^2 - 4AC for the conic 4x^2 - 4xy + y^2 = 0 and classify the conic.
-16, hyperbola
16, hyperbola
16, ellipse
0, parabola
For Ax^2 + Bxy + Cy^2, B^2 - 4AC = (-4)^2 - 441 = 16 - 16 = 0, which indicates a parabola. See Wikipedia.
Find the angle of rotation ? needed to eliminate the xy-term in the equation x^2 + 2?3 xy + 3y^2 = 6.
45
30
60
15
Using tan(2?) = B/(A - C) = (2?3)/(1 - 3) = -?3, so 2? = -60, giving ? = 30 in magnitude. See Lamar University Tutorial.
Determine the eigenvalues of the quadratic form Q(x,y) = 3x^2 - 6xy + 3y^2.
6 and 0
1 and 5
3 and 3
-3 and 9
The associated matrix is [[3, -3],[-3,3]]. Its eigenvalues are ? = 6 and 0, found by solving det|A - ?I| = 0. See MathWorld.
Which of the following is a valid parametric representation of the hyperbola x^2/16 - y^2/9 = 1?
x = 3 cosh t, y = 4 sinh t
x = 4 e^t, y = 3 e^{-t}
x = 4 cosh t, y = 3 sinh t
x = 4 sec t, y = 3 tan t
The identity cosh^2 t - sinh^2 t = 1 leads directly to x = a cosh t, y = b sinh t for x^2/a^2 - y^2/b^2 = 1. See Wikipedia.
Given the ellipse x^2/25 + y^2/9 = 1, what is the eccentric angle t for the point (3, 2)?
arctan(2/3)
acos(2/3)
acos(3/5)
asin(3/5)
Parametric form x = 5 cos t, y = 3 sin t gives cos t = 3/5. Thus t = arccos(3/5). See MathWorld.
For the general second-degree equation x^2 - y^2 + 2xy = 1, what are the eigenvalues of its quadratic form matrix and what does this imply about the conic?
0 and 1, parabola
2 and -2, hyperbola
?2 and -?2, hyperbola
1 and 1, ellipse
The matrix [[1,1],[1,-1]] has eigenvalues ?2. One positive and one negative eigenvalue indicate a hyperbola. See Wikipedia.
Which condition on the coefficients A, B, and C in the general second-degree equation Ax^2 + Bxy + Cy^2 + = 0 indicates that the conic is a hyperbola?
B^2 - 4AC < 0
B^2 - 4AC > 0
B = 0
B^2 - 4AC = 0
A positive discriminant (B^2 - 4AC > 0) is the hallmark of a hyperbola in the general quadratic equation. See Math is Fun.
What is the equation of the rectangular hyperbola xy = 1 in standard form after rotating the axes by 45?
u^2 - 2v^2 = 1
u^2 - v^2 = 2
u^2 - v^2 = 1
u^2 + v^2 = 1
Using x=(u?v)/?2, y=(u+v)/?2 gives xy = (u^2 - v^2)/2. Setting that equal to 1 yields u^2 - v^2 = 2. See Lamar University Tutorial.
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Study Outcomes
Identify Standard Forms of Conic Sections -
Recognize and write the standard equations for circles, ellipses, parabolas, and hyperbolas to excel in the conics test.
Graph Conic Curves -
Plot circles, ellipses, parabolas, and hyperbolas by locating vertices, foci, axes, and asymptotes with precision.
Calculate Key Parameters -
Compute radii, focal lengths, eccentricity, and focal chords to solve quiz on conic sections problems accurately.
Distinguish Between Conic Types -
Analyze general second-degree equations to classify them correctly as circles, ellipses, parabolas, or hyperbolas.
Apply Conic Concepts to Problem Solving -
Use your understanding of conic curves to interpret and resolve real-world and theoretical scenarios in the conic sections quiz.
Cheat Sheet
Classification by Eccentricity -
Every conic can be identified by its eccentricity e: e=0 gives a circle, 0<e<1 an ellipse, e=1 a parabola, and e>1 a hyperbola. A handy mnemonic is "Circle's e is zero, ellipses stay below one, parabola equals one, hyperbola goes beyond!" (Source: MIT OpenCourseWare).
Standard Equations and Parameters -
Review the standard forms: (x - h)²/a²+(y - k)²/b²=1 for ellipses, y - k=1/(4p)(x - h)² for parabolas, and (x - h)²/a² - (y - k)²/b²=1 for hyperbolas, with center (h,k). Remember that a and b set the shape and size, while p in parabolas indicates focus distance (Source: Khan Academy).
Focus - Directrix Property -
Every point P on a conic satisfies PF/PD=e, where F is the focus, D is the directrix, and e the eccentricity. For ellipses and hyperbolas, e=c/a relates foci distance c to semi-major axis a; for parabolas, e=1 so the focal length p is key (Source: University of Cambridge).
Reflective and Real-World Applications -
Parabolas focus parallel rays to their focus (satellite dishes) while ellipses bounce rays between foci (whispering galleries). Hyperbolas reflect as if emanating from alternate foci - understand these for physics or engineering examples (Source: American Mathematical Society).
Graph Transformations and Asymptotes -
Practice horizontal/vertical shifts and rotations: moving (h,k) changes a conic's center, while rotate terms like xy produce tilted graphs. For hyperbolas, asymptotes y - k=±(b/a)(x - h) guide sketching - key for any practice conic test or conic sections quiz (Source: University of California).
