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:
- : This represents a circle centered at the origin with a radius of 6.
- : This represents a straight line with slope , meeting the x-axis at and the y-axis at .
Key Features to Identify:
- Circle: Look for a graph with a circle of radius 6 centered at the origin.
- Line: Look for a line that intersects the circle at two points and has slope , 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 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:
- How do you derive the points of intersection between the line and the circle?
- What are the general forms of the equations for circles and lines?
- How do you calculate the slope of a line given two points?
- Can a line intersect a circle at more than two points? Why or why not?
- 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
Related Recommendation
System of Equations: Circle x^2 + y^2 = 25 and Line x + y = 3
Find Slope and Intercept from the Circle Equation x^2 + y^2 = 3x
Find the Intersection of a Circle and a Line: (x - 7)^2 + (y + 2)^2 = 49, y = x - 2
Graphing Nonlinear Systems: Circle and Line Intersections
Sketching the Circle: Equation x^2 + (y - 6)^2 = 36