Parabola

Step 1: Find the vertex of the parabola
The vertex of a parabola is the midpoint between the focus and the directrix.
Given focus:
Given directrix:
The vertex is:

Step 2: Find the equation of the parabola
The standard equation of a parabola with vertex and focus (since it opens rightward) is: Here, , , and distance from vertex to focus: So, the equation becomes:

Final Answer: The equation of the parabola is .

Step 1: Identify the region of interest: - Above the parabola - Inside the circle - Bounded by the y-axis ( ) - In the first quadrant

Step 2: Find intersection points: 1. Between parabola and circle: Let : (only valid) (in first quadrant)

2. Between circle and y-axis:

Step 3: Set up the integral for the area: The area can be calculated as:

Step 4: Calculate each part: 1. Area under circle: This is a quarter-circle area minus a sector:

2. Area under parabola:

Step 5: Compute net area:

The set of points of the form ,

where is a real number, represents a parabola.

Let's analyze the given parametric equations:

These are parametric equations for the and coordinates of a point on the plane, where is the parameter.

We can eliminate the parameter to

Now can express in terms of , (which will help us identify the geometric shape of the curve.)

Hence we get:

Now, solving for t in terms of x

Substitute the expression for into

⇒

⇒

Now, we have an equation that relates x and y without any parameter t. The equation is a second-degree equation, which represents a parabola.

the vertex of the parabola represented by the equation is at the point .

The standard form of a quadratic equation is: where , , and . The formula for the vertex is: Now, substituting in the given equation: Thus, the vertex is .
Final Answer: