Asked by Mehul Mundhada on Apr 24, 2024
Verified
Write the standard form of the equation of the parabola with focus (0,-2) and its vertex at the origin.
A) y2=8yy ^ { 2 } = 8 yy2=8y
B) x2=8xx ^ { 2 } = 8 xx2=8x
C) y2=−8xy ^ { 2 } = - 8 xy2=−8x
D) x2=−8yx ^ { 2 } = - 8 yx2=−8y
E) x2=18yx ^ { 2 } = \frac { 1 } { 8 } yx2=81y
Parabola
A symmetrical, open curve that represents the graph of a quadratic function, defined as the set of all points equidistant from a point called the focus and a fixed line called the directrix.
Focus
A specific point used in the definition of conic sections, such as ellipses and hyperbolas, that helps determine their shape.
Vertex
A point where two or more curves, edges, or lines meet; in the context of geometry, it is the corner point of a polygon or polyhedron.
- Absorb the conventional expressions of parabola equations and their correlation with the focus and vertex.
Verified Answer
AS
Annmaria Subash5 days ago
Final Answer :
D
Explanation :
The standard form of the equation of a parabola with vertex at the origin and focus at (0, p) or (p, 0) is given by:
4p(y2−x2)=x2+y24p(y^2-x^2)=x^2+y^24p(y2−x2)=x2+y2
In this case, the vertex is at the origin, so p = -2 (since the focus is at (0, -2)). Substituting p = -2, we get:
4(−2)(y2−x2)=x2+y24(-2)(y^2-x^2)=x^2+y^24(−2)(y2−x2)=x2+y2
Simplifying, we get:
−8y2+8x2=x2+y2-8y^2+8x^2=x^2+y^2−8y2+8x2=x2+y2
7x2+9y2=07x^2+9y^2=07x2+9y2=0
Dividing both sides by 9, we get:
x2−97+y217=1\frac{x^2}{-\frac{9}{7}}+\frac{y^2}{\frac{1}{7}}=1−79x2+71y2=1
Therefore, the standard form of the equation of the parabola is:
x2=−8yx^2 = -8yx2=−8y which is option D.
4p(y2−x2)=x2+y24p(y^2-x^2)=x^2+y^24p(y2−x2)=x2+y2
In this case, the vertex is at the origin, so p = -2 (since the focus is at (0, -2)). Substituting p = -2, we get:
4(−2)(y2−x2)=x2+y24(-2)(y^2-x^2)=x^2+y^24(−2)(y2−x2)=x2+y2
Simplifying, we get:
−8y2+8x2=x2+y2-8y^2+8x^2=x^2+y^2−8y2+8x2=x2+y2
7x2+9y2=07x^2+9y^2=07x2+9y2=0
Dividing both sides by 9, we get:
x2−97+y217=1\frac{x^2}{-\frac{9}{7}}+\frac{y^2}{\frac{1}{7}}=1−79x2+71y2=1
Therefore, the standard form of the equation of the parabola is:
x2=−8yx^2 = -8yx2=−8y which is option D.
Learning Objectives
- Absorb the conventional expressions of parabola equations and their correlation with the focus and vertex.