Pregunta
upstudy study bank question image url

Find the equation of a circle with diameter end points \( (-1,-2) \) and \( (3,2) \) \( \begin{array}{l}\text { [4 mks] } \\ \left.d=\sqrt{\left(-1-(3)^{2}+(-2\right.}-2\right)^{2} \\ d=\sqrt{16+2}\end{array} \)

Ask by Morgan Schultz. in Papua New Guinea
Mar 15,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The equation of the circle is \( (x-1)^2 + y^2 = 8 \).

Solución

¡Inicia sesión para desbloquear respuestas gratis!

Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

star-icon Descubrir

Respondido por UpStudy AI y revisado por un tutor profesional

Bonus Knowledge

To find the equation of the circle with the given diameter endpoints, we first calculate the center of the circle, which is the midpoint of the two endpoints. The midpoint \((h, k)\) can be calculated as: \[ h = \frac{-1 + 3}{2} = 1, \quad k = \frac{-2 + 2}{2} = 0. \] So the center of the circle is \((1, 0)\). Next, we find the radius \(r\) by calculating the distance between the center and one of the endpoints. Using the distance formula, we have: \[ r = \sqrt{(1 - (-1))^2 + (0 - (-2))^2} = \sqrt{(1 + 1)^2 + (0 + 2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}. \] The standard equation of a circle is given by: \[ (x - h)^2 + (y - k)^2 = r^2. \] Substituting \(h = 1\), \(k = 0\), and \(r = 2\sqrt{2}\): \[ (x - 1)^2 + (y - 0)^2 = (2\sqrt{2})^2. \] This simplifies to: \[ (x - 1)^2 + y^2 = 8. \] Therefore, the equation of the circle is: \[ \boxed{(x - 1)^2 + y^2 = 8}. \] Now, let's add some fun background information and applications of circles! A circle has been a part of human history for thousands of years! It’s fascinating to think that the ancient Greeks investigated the properties of circles long before calculators were invented. The simple beauty of a circle—where every point on the perimeter is equidistant from the center—has inspired art, architecture, and mathematics for centuries. Even today, the shape appears in everything from the design of bicycles to the wheels of cars! Circles aren't just mathematical constructs; they have practical applications in real life too! Think about circular tracks in sports, the gears in machines that rotate, or roundabout intersections that help manage traffic flow. Engineers use circles when designing anything from bridges to roller coasters. Plus, they're key in tech—as you move your mouse in a circular motion, you're controlling the movement of your cursor on a screen! So next time you see a circle, remember its rich history and its important role in the world around us!

preguntas relacionadas

Latest Geometry Questions

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad