A parabola is defined by the curve y² - 6y - 4x + 1 = 0. (i) Show that the curve is of the form (y - k)² = 4c(x - h). (ii) Identify V, the vertex of the parabola and its axis of sy... A parabola is defined by the curve y² - 6y - 4x + 1 = 0. (i) Show that the curve is of the form (y - k)² = 4c(x - h). (ii) Identify V, the vertex of the parabola and its axis of symmetry. (iii) State the coordinates of focus F. (iv) Sketch the parabola and locate the directrix.
Understand the Problem
The question involves solving a mathematical problem related to the properties and graphing of a parabola defined by a specific equation. It asks for multiple tasks, including showing a specific form of the equation, identifying features such as the vertex and focus, and sketching the parabola.
Answer
The vertex is $V(-2, 3)$, focus is $F(0, 3)$, directrix is $x = -4$. The parabola's equation is $(y - 3)^2 = 4(2)(x + 2)$.
Answer for screen readers
The equation of the parabola in the desired form is $(y - 3)^2 = 4(2)(x + 2)$.
- Vertex ( V: (-2, 3) )
- Focus ( F: (0, 3) )
- Directrix: ( x = -4 )
Steps to Solve
- Rearranging the Equation
Start with the given equation:
$$ y^2 - 6y - 4x + 1 = 0 $$
Rearranging it gives:
$$ y^2 - 6y = 4x - 1 $$
- Completing the Square for y
To express the left side as a perfect square, complete the square:
$$ y^2 - 6y = (y - 3)^2 - 9 $$
Substituting this back into the equation results in:
$$ (y - 3)^2 - 9 = 4x - 1 $$
- Final Form of the Equation
Rearranging gives:
$$ (y - 3)^2 = 4x + 8 $$
This can be written as:
$$ (y - 3)^2 = 4(2)(x + 2) $$
Thus, we have:
$$(y - k)^2 = 4c(x - h)$$
Where $k = 3$, $h = -2$, and $c = 2$.
- Identifying Vertex and Axis of Symmetry
The vertex ( V ) of the parabola is at the point ( (h, k) = (-2, 3) ). The axis of symmetry is the line ( y = 3 ).
- Finding the Focus
The focus ( F ) is located at ( (h + c, k) = (-2 + 2, 3) = (0, 3) ).
- Locating the Directrix
The equation of the directrix is given by ( x = h - c ):
$$ x = -2 - 2 = -4 $$
- Sketching the Parabola
The parabola opens to the right, with vertex at ( (-2, 3) ), a focus at ( (0, 3) ), and a directrix at ( x = -4 ).
The equation of the parabola in the desired form is $(y - 3)^2 = 4(2)(x + 2)$.
- Vertex ( V: (-2, 3) )
- Focus ( F: (0, 3) )
- Directrix: ( x = -4 )
More Information
This form indicates that the parabola opens to the right. The focus is always on the axis of symmetry, and the directrix is perpendicular to the axis of symmetry, serving as a guide for the shape of the parabola.
Tips
- Failing to complete the square correctly, which can result in an incorrect vertex or equation form.
- Confusing the positions of the focus and the directrix.
- Misidentifying the direction of the parabola based on its equation.
AI-generated content may contain errors. Please verify critical information
