To determine the coordinates of the vertex, focus, equation of the directrix, and the length of the latus rectum of the given parabola, we can follow these steps:

Step 1: Rewrite the equation of the parabola in standard form. The given equation is 4y^2 + 3x - 8y - 2 = 0. To rewrite it in standard form, we need to complete the square for the y terms.

Rearranging the equation, we have:
4y^2 - 8y = -3x + 2

Factoring out 4 from the y terms, we get:
4(y^2 - 2y) = -3x + 2

Completing the square inside the parentheses, we add (2/2)^2 = 1 to both sides:
4(y^2 - 2y + 1) = -3x + 2 + 4

Simplifying further, we have:
4(y - 1)^2 = -3x + 6

Step 2: Identify the values of h and k. The standard form of a parabola is (y - k)^2 = 4a(x - h), where (h, k) represents the coordinates of the vertex. Comparing this with our equation, we can see that h = 0 and k = 1.

Step 3: Find the coordinates of the vertex. Since h = 0 and k = 1, the vertex is located at (0, 1).

Step 4: Determine the value of a. The value of a can be found by dividing the coefficient of x by 4a in the standard form equation. In our case, 4a = -3, so a = -3/4.

Step 5: Calculate the coordinates of the focus. The focus of a parabola is given by (h, k + 1/4a). Substituting the values, we have:
Focus = (0, 1 + 1/(-3/4)) = (0, 1 - 4/3) = (0, -1/3)

Step 6: Determine the equation of the directrix. The equation of the directrix for a parabola is given by y = k - 1/4a. Substituting the values, we have:
Directrix = y = 1 - 1/(-3/4) = 1 + 4/3 = 7/3

Step 7: Find the length of the latus rectum. The length of the latus rectum is given by 4a. In our case, 4a = 4(-3/4) = -3.

Step 8: Sketch the curve. Using the information obtained, we can plot the vertex at (0, 1), the focus at (0, -1/3), and the directrix at y = 7/3. The latus rectum has a length of -3.

By following these steps, we can determine the coordinates of the vertex, focus, equation of the directrix, and the length of the latus rectum, and sketch the curve of the given parabola.

Question

To determine the coordinates of the vertex, focus, equation of the directrix, and the length of the latus rectum of the given parabola, we can follow these steps:

Step 1: Rewrite the equation of the parabola in standard form. The given equation is 4y^2 + 3x - 8y - 2 = 0. To rewrite it in standard form, we need to complete the square for the y terms.

Rearranging the equation, we have:
4y^2 - 8y = -3x + 2

Factoring out 4 from the y terms, we get:
4(y^2 - 2y) = -3x + 2

Completing the square inside the parentheses, we add (2/2)^2 = 1 to both sides:
4(y^2 - 2y + 1) = -3x + 2 + 4

Simplifying further, we have:
4(y - 1)^2 = -3x + 6

Step 2: Identify the values of h and k. The standard form of a parabola is (y - k)^2 = 4a(x - h), where (h, k) represents the coordinates of the vertex. Comparing this with our equation, we can see that h = 0 and k = 1.

Step 3: Find the coordinates of the vertex. Since h = 0 and k = 1, the vertex is located at (0, 1).

Step 4: Determine the value of a. The value of a can be found by dividing the coefficient of x by 4a in the standard form equation. In our case, 4a = -3, so a = -3/4.

Step 5: Calculate the coordinates of the focus. The focus of a parabola is given by (h, k + 1/4a). Substituting the values, we have:
Focus = (0, 1 + 1/(-3/4)) = (0, 1 - 4/3) = (0, -1/3)

Step 6: Determine the equation of the directrix. The equation of the directrix for a parabola is given by y = k - 1/4a. Substituting the values, we have:
Directrix = y = 1 - 1/(-3/4) = 1 + 4/3 = 7/3

Step 7: Find the length of the latus rectum. The length of the latus rectum is given by 4a. In our case, 4a = 4(-3/4) = -3.

Step 8: Sketch the curve. Using the information obtained, we can plot the vertex at (0, 1), the focus at (0, -1/3), and the directrix at y = 7/3. The latus rectum has a length of -3.

By following these steps, we can determine the coordinates of the vertex, focus, equation of the directrix, and the length of the latus rectum, and sketch the curve of the given parabola.

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