The distance between two points in three-dimensional space can be found using the following formula: d = √((x2 – x1)^2 + (y2 – y1)^2 + (z2 – z1)^2) where (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points, and d is the distance between them. To use the formula, you…
Locus problems are mathematical problems that involve finding the set of all points that satisfy a given condition or set of conditions. The solution to a locus problem is the set of all points that satisfy the given conditions, which is often referred to as the locus of the problem. Locus problems can be two-dimensional…
The equation of the tangent and normal to a curve at a given point can be found using calculus. Given a curve with equation y = f(x), the derivative of the function with respect to x is dy/dx. The value of the derivative at a specific point (x0, y0) gives the slope of the tangent…
Parametric equations are a way to represent a set of equations with one or more parameters. In mathematics, they are often used to describe curves and surfaces in space. Parametric equations are usually written in the form: x = f(t) y = g(t) where t is the parameter, and f(t) and g(t) are functions that…
Directrices and eccentricity are terms commonly used in geometry to describe the properties of a conic section, which is a curve obtained by the intersection of a plane with a cone. These terms are particularly useful in understanding the properties of ellipses and hyperbolas. The directrices of an ellipse are two lines that are equidistant…
In analytical geometry, the foci are points that are used to define the shape of an ellipse or a hyperbola. For an ellipse, the foci are two fixed points inside the ellipse such that the sum of the distances from any point on the ellipse to the two foci is a constant. This constant is…
An ellipse is a geometric shape that looks like a flattened circle. It is defined as the set of all points in a plane, the sum of whose distances from two fixed points (called the foci) is constant. The standard form equation of an ellipse centered at the origin is: (x^2 / a^2) + (y^2…
The standard equation of a parabola is: y = a x^2 + b x + c where: The vertex form of a parabola is: y = a(x – h)^2 + k where: The focus and directrix of a parabola can also be expressed in terms of a, as follows: Note that the vertex form of…
To find the equation of a circle passing through the points of intersection of two circles, we can follow these steps: Let’s say the two circles have equations: (x – a)^2 + (y – b)^2 = r^2 (x – c)^2 + (y – d)^2 = s^2 where (a,b) and (c,d) are the centers of the…
The intersection of a circle and a straight line or a circle can take different forms depending on the relative positions and sizes of the objects involved. Intersection of a circle and a straight line: If a circle and a straight line lie in the same plane, they can intersect in three different ways: Intersection…
