PRE CALCULUS Conic Sections and Parametric Equations 1 — Questions and Answers
Question 1: What is the standard form of a circle with center (h, k) and radius r?
- (x−h)² + (y−k)² = r² (Correct answer)
- (x+h)² + (y+k)² = r
- x² + y² = r
- (x−h)/(y−k) = r²
Correct answer: (x−h)² + (y−k)² = r²
The standard equation of a circle is (x−h)² + (y−k)² = r², where (h, k) is the center.
Question 2: Which conic section is formed when the cutting plane is parallel to the base of a double cone?
- Circle (Correct answer)
- Parabola
- Ellipse
- Hyperbola
Correct answer: Circle
A horizontal cut parallel to the base of a cone produces a circle.
Question 3: The focus of the parabola y = x² is located at:
- (0, 1/4) (Correct answer)
- (0, 1)
- (1/4, 0)
- (0, −1/4)
Correct answer: (0, 1/4)
Rewriting y = x² as x² = y in form x² = 4py gives 4p = 1, so p = 1/4 and the focus is (0, 1/4).
Question 4: What distinguishes an ellipse from a hyperbola in standard form?
- Ellipse uses + between terms; hyperbola uses − (Correct answer)
- Ellipse uses − between terms; hyperbola uses +
- Both use + between terms
- The denominators are equal in a hyperbola
Correct answer: Ellipse uses + between terms; hyperbola uses −
The standard ellipse equation has the form x²/a² + y²/b² = 1, while the hyperbola uses a minus sign.
Question 5: In the ellipse x²/25 + y²/9 = 1, what is the length of the major axis?
- 10 (Correct answer)
- 5
- 9
- 6
Correct answer: 10
The major axis has length 2a where a² = 25, so a = 5 and the major axis = 10.
Question 6: A parametric equation describes:
- Both x and y as functions of a third variable t (Correct answer)
- y as a function of x directly
- Only circular motion
- A 3D surface
Correct answer: Both x and y as functions of a third variable t
Parametric equations express x and y each in terms of a parameter t, allowing curves that are not functions of x alone.
PRE CALCULUS Conic Sections and Parametric Equations 1
What is the standard form of a circle with center (h, k) and radius r?