Will you please check my work for me.
1)The equation of a parabola is shown.
y=1/14X^2. What are the coordinates of the focus?
(0,-3.5) <----
(0,-4)
(0,7)
2)The equation x2 + (y + 3)2 = 36 models the boundary on a local map for which Darren can hear his friend Tom on his two-way radio when Darren is at home. How far (in miles) can Tom walk from Darren's home and still be heard?
3 miles
6 miles<----
2 miles
12 miles
3)A plane intersects only one nappe of a double-napped cone. It is neither perpendicular to the cone's axis nor parallel to its generating line. Which conic section is formed?
point
circle
ellipse <-----
parabola
1) To find the coordinates of the focus of a parabola, you need to know the equation of the parabola in standard form, which is of the form y = ax^2. In this case, you have the equation y = (1/14)x^2.
The general equation of a parabola in standard form is given by:
y = 1/4a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola.
Comparing this general form to the given equation y = (1/14)x^2, we can see that a = 1/14, h = 0, and k = 0.
The focus of a parabola is a point located inside the parabola along its axis of symmetry, which is given by the equation x = h. Hence, in this case, the focus has coordinates (0, p), where p is calculated as p = k + 1/4a.
For the given equation, p = 0 + 1/(4 * 1/14) = 0 + 14/4 = 3.5.
Therefore, the coordinates of the focus are (0, -3.5).
2) The equation x^2 + (y + 3)^2 = 36 represents the boundary on the local map where Darren can hear his friend Tom on his two-way radio when Darren is at home.
The equation of a circle in standard form is given by:
(x - h)^2 + (y - k)^2 = r^2
Comparing this general form to the given equation x^2 + (y + 3)^2 = 36, we can see that the center of the circle is located at (h, k) = (0, -3) and the radius is given by r = √(36) = 6.
Therefore, Tom can walk up to a distance of 6 miles from Darren's home and still be heard.
3) A plane that intersects only one nappe of a double-napped cone, not perpendicular to the cone's axis nor parallel to its generating line, forms an ellipse.
An ellipse is a type of conic section that is formed when a plane intersects a cone at an angle and cuts through both halves of the cone, resulting in a closed curve that is symmetric around two points called the foci.
Therefore, the correct conic section formed by the given scenario is an ellipse.