In mathematics, a parabola is a plane curve, mirror-symmetrical and is U-shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define the same curves. One description of a parabola involves a line; the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane, tangential to the conical surface; the line perpendicular to the directrix and passing through the focus is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most curved; the distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola, parallel to the directrix and passes through the focus.

Parabolas can open up, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry; the same effects occur with other waves. This reflective property is the basis of many practical uses of parabolas; the parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. They are used in physics and many other areas; the earliest known work on conic sections was by Menaechmus in the 4th century BC.

He discovered a way to solve the problem of doubling the cube using parabolas. The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola; the name "parabola" is due to Apollonius. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved; the focus–directrix property of the parabola and other conic sections is due to Pappus. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity; the idea that a parabolic reflector could produce an image was well known before the invention of the reflecting telescope. Designs were proposed in the early to mid-17th century by many mathematicians, including René Descartes, Marin Mersenne, James Gregory; when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror.

Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as a set of points in the Euclidean plane: A parabola is a set of points, such that for any point P of the set the distance | P F | to a fixed point F, the focus, is equal to the distance | P l | to a fixed line l, the directrix:; the midpoint V of the perpendicular from the focus F onto the directrix l is called vertex, the line F V is the axis of symmetry of the parabola. If one introduces Cartesian coordinates, such that F =, f > 0, the directrix has the equation y = − f, one obtains for a point P = from | P F | 2 = | P l | 2 the equation x 2 + 2 = 2. Solving for y yields y = 1 4 f x 2; this parabola is U-shaped. The horizontal chord through the focus is called the latus rectum; the latus rectum is parallel to the directrix. The semi-latus rectum

Option Grid is the name for a tool for patients and providers to use together when they are discussing and deciding what best to do about possible options, either treatments or tests. The grid is published in the form of a summary table to enable comparisons between multiple potential treatments or options; the grids do this by using questions that patients ask, are designed for use in face-to-face clinical encounters or to be given to patients to read for a few minutes, ahead of a conversation with a provider. The key to the grids is the use of asked questions that relate to the most common or most important concerns of patients, it is important to choose these FAQs and to limit them to those that can be considered briefly. These FAQs are based on evidence where possible, final versions are developed by teams of patients and editors. All Grids are written at a reading level of 10–12 years, in accordance with the plain English campaign guides; the evidence summaries upon which Option Grids are based are available for public review at the official Option Grid website.

A number of option grids exist including: Lumpectomy with radiotherapy vs. mastectomy for breast cancer Surgery to remove the ovaries and fallopian tubes vs. no surgery for ovarian cancer risk before menopause Surgery to remove the ovaries and fallopian tubes vs. no surgery forovarian cancer risk after menopause Implant an implantable cardioverter defibrillator and continue to use medication vs. continue to use medication but do not implant an ICD for patients with heart failure Optimal medical treatment vs. angioplasty for stable angina Dabigatran vs. warfarin for stroke prevention with non-valvular atrial fibrillation Immunomodulator vs. anti-TNF vs. combination therapy for Crohn’s disease Continuing medical treatment without surgery vs. brain surgery, for hippocampal sclerosis in temporal lobe epilepsy Sodium valproate vs. Levetiracetam vs. Lamotrigine for epilepsy treatments when considering pregnancy Peritoneal dialysis vs. haemodialysis at the hospital vs. haemodialysis at home vs. transplantation vs. conservative management for chronic kidney disease Continuing to stay off paid work vs. taking steps to return to paid work with the help of an employment specialist Non-operative treatment vs. hip replacement surgery for osteoarthritis of the hip Painkillers vs. joint injections vs. knee replacement surgery for managing knee pain and activity level in osteoarthritis of the knee Lifestyle and weight loss vs. medication for self-management of osteoarthritis of the knee Managing without injections or surgery vs. injections vs. surgery for sciatica from a herniated disc Treatment with antibiotics vs. treatment without antibiotics for a sore throat Circumcision vs. no circumcision for newborn boy Grommets vs. hearing aid vs. active observation for glue ear in child Tonsillectomy vs. active management for tonsillitis in children under 16 Having amniocentesis vs. not having amniocentesis for testing for Down’s Syndrome in pregnancy Having Down syndrome screening vs. not having Down syndrome screening in pregnancy Medication vs. hormonal medication vs. removing the uterus lining vs. removing fibroids vs. removing the uterus for heavy menstrual bleeding The Option Grid Collaborative is a not-for-profit group of over 90 people, made up of patient representatives, medical experts, clinicians involved in supporting shared decision making via the creation of Option Grids.

The Collaborative welcomes new members who wish to create new Option Grids according to the group’s agreed-upon process, which involves a thorough review of best available evidence and user testing process. Interested members should visit the Official Website in the links below for more information about how to get involved. Collaborative members receive support and guidance for Option Grid development from Dartmouth College throughout the process; the Collaborative operates under a Creative Commons license, a type of public copyright license that enables authors to give others the right to collaborate and build upon their work according to guidelines specified by the author. The Collaborative’s specific license, CC-BY-NC-ND, allows others to download the group’s work and share it so long as they credit the source, do not make changes, do not use it commercially. Decision aids Shared decision making Patient participation Official website video example of how Option Grid might be used

Charles W. Murphy was an American investor and hedge fund manager and has been referred to in media reports as a "financial guru". Charles W. Murphy was born in July 1961 to a middle-class New York family, he attended The Buckley School and Stuyvesant High School. He graduated with a B. A. from Columbia University, where he was a member of the starting varsity crew team, earned a law degree from Harvard University, an MBA from the Massachusetts Institute of Technology. Murphy began his career at Goldman Sachs joined Morgan Stanley as a managing director in the Financial Institutions Group. After ten years at Morgan Stanley he moved to London where he led the Financial Institutions Group at Deutsche Bank and at Credit Suisse. Murphy worked at Fairfield Greenwich, which had invested more than $7 billion in the notorious Madoff Ponzi scheme. Murphy was unaware. Murphy joined investment firm Paulson & Co. in 2009. He rose to the level of Partner. While there, he led many successful investments, including help organize the firm's activist push to break up American International Group.

Murphy owned a 19th century, 11,550-square-foot home in New York that he purchased in 2007 for $33 million, a record price at the time, from Seagram heir Matthew Bronfman. In 2016, Murphy listed the property for sale for $49.5 million. In April 2018 Murphy's widow, Annabella Murphy, sold the seven story home for $28.5 million. At the time of his death, Murphy was married to his second wife Annabella Murphy and had four children. Murphy died on Monday March 28, 2017 at the age of 56 after falling, dressed in a suit, from the 24th floor of the Sofitel Hotel in midtown Manhattan. Three weeks before his death, Murphy added his wife, Annabella to the deed of their home, giving her sole ownership of the property, which suggests Murphy may have planned his death in advance. Murphy was being treated for depression by a psychiatrist and was taking anti-depressants at the time of his death