Question

In the standard \((x, y)\) coordinate plane, an equation of a circle is \(x^{2}+y^{2}=81 .\) At what points does the circle intersect the \(y\) -axis? A. \((0,1)\) and \((0,-1)\) B. \((0,9)\) and \((0,-9)\) C. \((0,18)\) and \((0,-18)\) D. \((0,27)\) and \((0,-27)\) E. \((0,81)\) and \((0,-81)\)

Short answer

Answer: The circle intersects with the y-axis at points (0, 9) and (0, -9).

Step-by-step solution

Plug in x = 0

Since we are looking for the points where the circle intersects the \(y\)-axis, we know that the \(x\)-coordinate is \(0\). So, we can plug \(x=0\) into the circle equation: \((0)^{2}+y^{2}=81\)

Solve for y

Simplify the equation to solve for \(y\): \(y^{2}=81\). To find the points of intersection, we must find the positive and negative values of \(y\) that satisfy this equation. Taking the square root of both sides, we get \(y = \pm9\).

Identify the Points of Intersection

Since we know the \(x\)-coordinate is \(0\) and the two possible \(y\)-coordinates are \(9\) and \(-9\), the points of intersection are \((0,9)\) and \((0,-9)\).

Match with the Provided Options

Compare our answer with the given options, and we can see that our answer matches option B: \((0,9)\) and \((0,-9)\). So, the circle intersects the \(y\)-axis at points \((0,9)\) and \((0,-9)\). The correct answer is \(\boxed{\text{B}}\).

Key concepts

Standard Coordinate Plane

Understanding the standard coordinate plane is crucial for visualizing and solving geometry problems. This plane is like a flat canvas that stretches infinitely in all directions, divided into four quadrants by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is identified by a pair of numbers, \((x, y)\), which represent its horizontal and vertical distances from the origin \((0, 0)\), the point where the x and y-axis intersect.

When dealing with circle equations, we visualize them as curves on this plane, with the center and the radius defined by the equation. The circle's intersection with the axes tells us a lot about its position and size. For example, in our exercise, since the circle's equation is \(x^{2}+y^{2}=81\), we know it has a radius \(9\), centered at the origin.

Solving for y

When we 'solve for y', we're trying to find the value of y in terms of other variables or numbers. In algebra, this often means isolating y on one side of an equation. This process is key to unlocking the mysteries of an equation and finding solutions that satisfy it.

Take the circle equation \(x^{2}+y^{2}=81\). When we're told that a circle intersects the y-axis, we know the x-value at those points is \(0\) since that's where the y-axis lies. By plugging in \(x = 0\) into the equation, the equation simplifies to \(y^{2} = 81\). We then solve for y by taking the square root of both sides, considering both the positive and negative roots, since the circle lies both above and below the x-axis.

Points of Intersection

A point of intersection is where two or more curves or lines meet on a graph. For circles on a coordinate plane, these can be points where the circle meets the x or y-axis, or any other line or curve. Determining these points can be crucial for understanding the properties and the position of geometrical shapes.

In the context of the circle equation from our exercise, finding the points of intersection with the y-axis simplifies to finding the y-values when \(x = 0\). Once we have solved the equation for y, we discover two solutions, \(y = 9\) and \(y = -9\), symbolizing the points (0, 9) and (0, -9) where our circle touches the y-axis.