Circle

We are given the equation of the circle and the equation of the chord . We are also told that is the midpoint of this chord. We are required to find the value of .
Step 1: First, we rewrite the equation of the circle in standard form by completing the square for both and .
Starting with the equation of the circle: Complete the square for and : - For , we complete the square by adding and subtracting .
- For , we complete the square by adding and subtracting .
The equation becomes: Thus, the center of the circle is , and the radius is .
Step 2: Next, we find the equation of the chord. The equation of the chord is , which we can rewrite as: Now, the midpoint of the chord is the point on the line that is equidistant from both ends of the chord.
Step 3: We use the property of the midpoint of a chord. For a circle with equation and a chord , the midpoint satisfies the relation: Substituting the values of into the equation, we get: So, the equation becomes: Step 4: Since is the midpoint of the chord, it lies on the line , and the equation for the circle is . Solving these equations, we obtain .
Thus, the value of is .

Step 1: Finding the center and radius of the given circle The given circle equation: Rewriting in standard form, complete the square: Thus, the center is and radius .
Step 2: Finding the required circle The required circle has radius and is internally tangent at . Using the equation transformation method and substituting , we determine . After calculation,

We are given the following information:
- and are the equations of two diameters of the circle.
- The equation of the circle is .
- The point lies outside the circle, and tangents are drawn from this point to the circle.
Step 1: We start by finding the center and radius of the circle using the general equation of the circle , where and . We need to determine the values of and from the equations of the diameters.
Step 2: The diameters and can be used to find the center of the circle as the intersection point of these two lines. We solve the system of linear equations: Solving for and , we multiply the first equation by 2 and subtract from the second equation: Subtracting these: Substitute into : Thus, the center of the circle is .
Step 3: Now, we can calculate the area of the quadrilateral .
The area of the quadrilateral formed by the points is given by the area formula for a polygon: Substituting the points into the formula: Thus, the area of the quadrilateral is .

Step 1: Understanding the given conditions The given lines and are perpendicular, forming a square with the given circle.
Step 2: Finding the required circle equation Using the standard form of a circle and solving for the appropriate radius satisfying the tangency conditions, we get: