Circle

The equation of the circle is . The center of the circle is , and the radius is . The point of tangency is . The tangent at this point is perpendicular to the radius at this point.
Step 1: Find the slope of the radius The slope of the radius is given by the formula for the slope between two points:
Step 2: Find the slope of the tangent The slope of the tangent line is the negative reciprocal of the slope of the radius. Therefore, the slope of the tangent is:
Step 3: Use the point-slope form of the line equation The point-slope form of a line is given by: where is the slope and is the point of tangency. Substituting and the point , we get: Simplifying:

Thus, the correct option is (A) :

The general equation of a circle is: where is the center of the circle and is the radius. 1. The center of the circle is on the line because the circle touches this line. So, the center has coordinates . 2. The circle touches the x-axis at , meaning the distance from the center to the x-axis is equal to the radius of the circle. This gives the equation: 3. The distance from the center of the circle to the point is also equal to the radius. Using the distance formula: Since , we get: Squaring both sides: Simplifying: Thus, . Now, we can write the equation of the circle with center and radius 10: Expanding: Simplifying:

The correct option is (A) :

Step 1: Identify the Given Circles

- First Circle: with center and radius .
- Second Circle: with center and radius (since ).

Step 2: Understand the Condition

The line segment joining points and on the first circle is the diameter of the second circle. This implies:

• The midpoint of and is the center of the second circle, .
• The distance between and must satisfy a specific geometric relationship with .

Step 3: Calculate the Distance Between Centers

Step 4: Determine the Radius

Since is the diameter of the second circle, its length is . The relationship between the distance between centers ( ) and the radius is given by:

The points lie on the boundary of a circle, implying that they satisfy the equation of a circle, i.e., the sum of squares of the coordinates equals 1. Therefore, we can write the relationship as:

Now, recalling the identity for secant: , we substitute this into the equation to get:

This simplifies to:

Multiplying through by gives:

Rearranging the terms, we find:

This equation can be solved numerically or through further algebraic manipulations depending on the properties of the angles .

Answer: The value of is , so the correct option is (D).

Let's rewrite the equation of the circle in the standard form , where is the center and is the radius. Given equation:

Complete the square for and terms: So the center is and the radius is .

1. The radius of C is : Correct

2. lies inside of C: Substitute into the equation: . Since the distance from the center is 0, which is less than the radius , lies inside the circle (it is the center). Correct

3. lies inside of C: Substitute into the equation: . Since the distance from the center is greater than the radius, lies outside the circle. Incorrect

4. The line intersects C: We can write . Substitute into the circle equation: or If , .
So, is an intersection point. If , . So, is an intersection point.
Since there are intersection points, the line intersects the circle. Correct

5. One of the diameters of C is not along : The equation can be written as . The line passes through the center since . Therefore, this line represents a diameter of the circle. Since there can be infinitely many diameters, one of them will not be along this line. Correct

The incorrect statement is lies inside of C. Final Answer:

The final answer is

Step 1: Find the sides of the square from the given equations

The square is formed by the lines:

• which factors to , giving vertical lines at and (width = units)
• which factors to , giving horizontal lines at and (height = units)

Thus, we have a square with side length 4 units.

Step 2: Determine the inscribed circle's properties

The circle inscribed in the square will:

• Have diameter equal to the side of the square, so diameter = 4 ⇒ radius
• Be centered at the intersection of the square's diagonals

Center coordinates: So the center is at .

Step 3: Write the equation of the circle

Using the standard form where is the center and is the radius: Expanding this: Rearranged form:

Rewriting the given circle equation into standard form: This shows that the center is at .
Since the center is the midpoint of the diameter, the other endpoint is determined using the midpoint formula: Solving for and , Thus, the other endpoint is .

The area of the segment of a circle is given by the formula: where: - (the radius of the circle), - , which is the central angle corresponding to the segment.
First, calculate the angle . The cosine inverse of gives .
Now, substitute the values of and into the formula for the area: This simplifies to: Thus, the area of the segment is .