Finding the Center and Radius of the Circle To find the center and radius of the circle given the equation x² + 2x + y² + 4y = 20, we need to rewrite this equation in the standard form of a circle, which is: (x – h)² + (y – k)² = r² where (h, … Read more
When the graph of a system of two linear equations has no solution, this indicates that the lines representing the equations are parallel. In more detail, parallel lines never intersect; thus, there is no point at which they meet. This lack of intersection implies that there are no values of the variables that satisfy both … Read more
The discriminant of a quadratic equation is a key component that helps determine the nature of the roots of the equation. For a standard quadratic equation of the form ax² + bx + c = 0, the discriminant can be calculated using the formula: D = b² – 4ac In your case, the quadratic equation … Read more
To derive the equation of a parabola from a given focus and directrix, we follow a geometric definition: a parabola is the set of all points (x, y) that are equidistant from the focus and the directrix. In this case, the focus is at the point (0, 4), and the directrix is given by the … Read more
To solve for the value of md, we first need to recognize that vertical angles are equal. This means that: mc = md According to the problem, we have: mc = 26x md = 2x + 10 Setting these two expressions equal to each other, we can write: 26x = 2x + 10 Now, let’s … Read more
To write an equation in point-slope form, you can use the formula: y – y1 = m(x – x1) Where: (x1, y1) is a point on the line (in this case, (8, 3)) m is the slope of the line (which is 8 here) Now, we can substitute the values into the formula: Point: (8, … Read more
The expression i0 + i1 + i2 + i3 + i4 involves the imaginary unit i, which is defined as the square root of -1. To evaluate this expression, we will calculate each power of i and then sum them up. Let’s break it down: i0 = 1 (Any number to the power of zero … Read more
The given expression appears to contain some formatting issues. However, if we interpret it as a standard quadratic equation, we can rewrite it more clearly. The standard form of a quadratic equation is typically stated as: ax² + bx + c = 0 In our case, it looks like you’re asking about the quadratic equation … Read more
Calculating Probabilities with a Six-Sided Die A standard six-sided die has six faces, numbered from 1 to 6. To find probabilities for specific scenarios, we can use the formula: Probability (P) = Number of favorable outcomes / Total number of possible outcomes Scenario A: Rolling a 5 or a Number Greater than 3 First, let’s … Read more
To find the area of the surface of the part of the plane defined by the equation 2xy + 3z = 1 that lies within the cylinder described by x^2 + y^2 = 3, we can follow these steps: Rearranging the Plane Equation: First, we need to express the plane equation in terms of z, … Read more