answer:1. **Key Properties and Theorems**: - **Power of a Point Theorem**: BC cdot AB = BD^2. - **Intersecting Chord Theorem**: AD cdot DC = BD cdot DA. 2. **Apply Power of a Point Theorem**: - Let BD = b and AD = d. Since AB = 2(2a) = 4a: [ BC cdot 4a = b^2. ] - Solving for BC: [ BC = frac{b^2}{4a}. ] 3. **Using Condition AE = EC** and AD = 2d (as it is double due to AE = EC): - AE = frac{1}{2} AC = frac{1}{2} (AD + DC). - Since DC = BC - BD, and substituting for BC: [ AE = frac{1}{2} (AD + BC - BD) = frac{1}{2} (2d + frac{b^2}{4a} - b). ] 4. **Relate x and y**: - By definition, x is AE: [ x = frac{1}{2} (2d + frac{b^2}{4a} - b). ] - y is the radius minus the perpendicular from E to AB, giving y = 2a. 5. **Formulate Equation**: - Express b in terms of x: [ b = 2a - x. ] - Substitute back to find x: [ x = frac{1}{2} (2d + frac{(2a-x)^2}{4a} - (2a - x)). ] - Solve for y^2: [ y^2 = (2a)^2 = 4a^2 = frac{x^3}{2a+x}. ] Thus, the correct relationship between x and y in this modified scenario is: [ y^2 = frac{x^3{2a+x}} ] The final answer is boxed{B}

answer:1. **Problem Setup:** We are given a quadrilateral (ABCD) where the diagonals (AC) and (BD) are perpendicular to each other. It is also given that a circle can be inscribed within the quadrilateral and a circle can be circumscribed around the quadrilateral. 2. **Objective:** We need to determine if the quadrilateral (ABCD) is always a square. 3. **Analysis:** We start by noting that the perpendicular diagonals suggest a specific property: - Let's consider a circle with diameter (AC). - Draw the chord (BD) which is perpendicular to (AC) but does not pass through the center (see reference diagram). 4. **Properties of the Perpendicular Chord:** Given the perpendicularity, the diameter (which is (AC)) divides this chord (BD) into two equal segments. 5. **Forming Right Triangles:** Since (AC) is a diameter, and it perpendicularly bisects (BD), we can observe: - In the triangles formed (ABD) and (BCD), the line (AC) serves as the hypotenuse (AB = AD) and (BC = CD). 6. **Isosceles Triangles:** The perpendicular bisector guarantees that triangles (ABD) and (BCD) are isosceles: [ AB = AD quad text{and} quad BC = CD ] 7. **Sums of Opposite Sides:** Development of the opposite sides yields: [ AB + CD = AD + BC ] This condition necessitates the possibility of an inscribed circle. 8. **Construction Validity:** Since the sum of the opposite sides of quadrilateral (ABCD) meets the necessary condition for it to inscribe a circle, this construction proves (ABCD) satisfies the given problem conditions. 9. **Conclusion:** Nonetheless, without additional constraints proving specifically that all sides are equal, we cannot definitively state (ABCD) is a square. It could be a kite shape or some other quadrilateral meeting the given conditions. Therefore, it is **not** possible to assert that the quadrilateral is necessarily a square. [ boxed{text{No, it cannot be definitively said that it is a square.}} ]

answer:To find the value of the number, you can set up an equation where the number divided by 6 equals 26: number / 6 = 26 To solve for the number, multiply both sides of the equation by 6: number = 26 * 6 number = 156 So the value of the number is boxed{156} .

answer:First, we need to find the inverse function of f. Let's set f(y) = x implies frac{y-5}{y-6} = x. To solve for y, multiply both sides by y-6: y-5 = x(y-6). Expanding and rearranging terms gives: y - xy = 5 - 6x implies y(1-x) = 5 - 6x. Assuming x neq 1, isolate y: y = frac{5 - 6x}{1 - x}. This expression gives us f^{-1}(x). f^{-1}(x) gets undefined where the denominator is zero. Thus, set 1-x = 0, x = 1. Conclusion: f^{-1}(x) is undefined for boxed{x = 1}.