Question

For nonzero constants \(a\) and \(b\), find the intercepts of the graph of $$ \left(x^{2}+y^{2}-a x\right)^{2}=b^{2}\left(x^{2}+y^{2}\right) $$ Then test for symmetry with respect to the \(x\) -axis, the \(y\) -axis, and the origin.

Short answer

The x-intercepts are found by setting y=0 and solving for x. The y-intercepts are found by setting x=0 and solving for y. To check for symmetry, replace x and y accordingly as per the axis being tested.

Step-by-step solution

Finding the x-intercepts

To find the x-intercepts, set y = 0 in the equation and solve for x. This gives: . Simplifying this equation, we get:

Solving for x

Solving the simplified equation: This reveals the possible x-intercepts.

Finding the y-intercepts

To find the y-intercepts, set x = 0 and solve for y. This gives: Simplifying, we find: This reveals the possible y-intercepts.

Testing for Symmetry with respect to the x-axis

To test for symmetry with respect to the x-axis, replace y with -y in the original equation and check if the equation remains unchanged. Compare and see if both sides are equal.

Testing for Symmetry with respect to the y-axis

To test for symmetry with respect to the y-axis, replace x with -x in the original equation and check if the equation remains unchanged. Compare and see if both sides are equal.

Testing for Symmetry with respect to the Origin

To test for symmetry with respect to the origin, replace x with -x and y with -y in the original equation and check if the equation remains unchanged. Compare and see if both sides are equal.

Key concepts

x-intercepts

In coordinate geometry, x-intercepts are the points where the graph of an equation crosses the x-axis. These points occur where the value of y is zero. To find the x-intercepts of an equation, substitute y = 0 and solve for x.
For the given equation \( (x^2 + y^2 - a{x})^2 = b^2{(x^2 + y^2)} \) set y = 0, leading to \( (x^2 - a{x})^2 = b^2 x^2 \). Simplifying further, you can solve for the values of x that make this equation true. These solutions are your x-intercepts.

• Example: y=0 in the equation \ x - 5 = 0 \ we solve for x. Here, the x-intercept is x = 5.

y-intercepts

Y-intercepts, on the other hand, are points where the graph crosses the y-axis. These points occur where x is zero. To identify y-intercepts, you substitute x = 0 in the equation and solve for y.
For the given equation \( (x^2 + y^2 - a{x})^2 = b^2{(x^2 + y^2)} \) set x = 0, which simplifies to \( y^4 = b^2 y^2 \). Further simplification reveals the values of y that satisfy this equation. These values are your y-intercepts.

• Example: x=0 in the equation \ 2y-6=0 \ we solve for y. Here, the y-intercept is y = 3.

graph symmetry

Graph symmetry helps identify whether the graph remains unchanged under certain transformations. There are three types of symmetry usually checked:

• **X-axis symmetry**: Replace y with -y in the equation and check if the equation stays the same.
• **Y-axis symmetry**: Replace x with -x in the equation and see if it remains unchanged.
• **Origin symmetry**: Replace x with -x and y with -y.

For our equation, you need to perform each transformation, simplify, and compare to check for symmetry.

algebraic equations

Understanding algebraic equations is crucial for finding intercepts and testing graph symmetry. An algebraic equation involves variables and constants combined with operations like addition, subtraction, multiplication, and powers. Solving these equations requires understanding:

• **Algebraic manipulation**: Simplifying expressions to find intercepts or test symmetry.
• **Substitution**: Replacing variables with specific values (like zero) to find intercepts or test transformations.

Through consistent practice, handling equations becomes an easier task.

coordinate geometry

Coordinate geometry combines algebra and geometry to analyze geometric properties within a coordinate system. It visualizes algebraic equations as graphs on the coordinate plane. Key concepts include:

• **Points and coordinates (x, y)**: Identifying positions on the graph.
• **Intercepts**: Points where the graph crosses the axes.
• **Graph transformations**: Moving, reflecting, or rotating graphs.

Mastering these concepts aids in better understanding the relationships and properties of geometric figures represented algebraically.