Math Problem Statement

Choose the graph that matches the system of equations. xsquaredplusysquaredequals36 xplusyequals3 Question content area bottom Part 1 Choose the correct graph. A. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 10 to 10 in increments of 2. An ellipse centered at the origin has its major axis horizontal and meets the x-axis at negative 6 and 6. Its minor axis is half the length of its major axis. A line with slope 1 rises from left to right and intersects the ellipse in two points. The line meets the x-axis at negative 3 and the y-axis at 3. B. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 10 to 10 in increments of 2. A circle of radius 6 is centered at the origin. A line falls from left to right and has slope negative 1. It meets the x-axis at 3 and the y-axis at 3. It intersects the circle in quadrants 2 and 4. C. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 10 to 10 in increments of 2. A parabola opens upward and has vertex at (0, negative 6). A line with slope negative 1 falls from left to right and intersects the parabola at two points. It meets the x-axis at 3 and the y-axis at 3. D. -10 10 -10 10 x y

Solution

Let’s analyze the given equations step by step to match the system with the correct graph:

System of Equations:

  1. x2+y2=36x^2 + y^2 = 36: This represents a circle centered at the origin (0,0)(0, 0) with a radius of 6.
  2. x+y=3x + y = 3: This represents a straight line with slope 1-1, meeting the x-axis at (3,0)(3, 0) and the y-axis at (0,3)(0, 3).

Key Features to Identify:

  1. Circle: Look for a graph with a circle of radius 6 centered at the origin.
  2. Line: Look for a line that intersects the circle at two points and has slope 1-1, cutting through quadrants 2 and 4.

Evaluate the Options:

  • Option A: Describes an ellipse, not a circle. Incorrect.
  • Option B: Describes a circle of radius 6 centered at the origin and a line with slope 1-1 intersecting the circle in quadrants 2 and 4. Matches our system perfectly.
  • Option C: Describes a parabola, not a circle. Incorrect.
  • Option D: Not described in detail but seems unnecessary to analyze since B is correct.

Correct Answer:

B

Would you like a detailed breakdown of why the other options don't work? Let me know!


Related Questions:

  1. How do you derive the points of intersection between the line and the circle?
  2. What are the general forms of the equations for circles and lines?
  3. How do you calculate the slope of a line given two points?
  4. Can a line intersect a circle at more than two points? Why or why not?
  5. How do you verify that a given point lies on both the circle and the line?

Tip:

Always sketch the graphs of equations to visually confirm solutions when solving systems involving geometry.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Equations of Circles
Linear Equations

Formulas

Equation of a circle: x^2 + y^2 = r^2
Equation of a line: y = mx + b

Theorems

A line can intersect a circle at most at two points.
The slope-intercept form of a line defines its slope and intercepts.

Suitable Grade Level

Grades 9-12