Problem 45 - Algebra and Trigonometry

Question

In the derivation of the equation of the hyperbola at the beginning of this section, we said that the equation $$ \sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=\pm 2 a $$ simplifies to $$ \left(c^{2}-a^{2}\right) x^{2}-a^{2} y^{2}=a^{2}\left(c^{2}-a^{2}\right) $$ Supply the steps needed to show this.

Step-by-Step Solution

Answer

The equation simplifies to the hyperbola form through squaring, expanding, and simplifying.

Step 1: Square Both Sides of the Equation

We begin by squaring both sides of the original equation to eliminate the square roots. The original equation is: \[ \sqrt{(x+c)^{2}+y^{2}} - \sqrt{(x-c)^{2}+y^{2}} = \pm 2a \] Squaring both sides gives us: \[ \left( \sqrt{(x+c)^{2}+y^{2}} - \sqrt{(x-c)^{2}+y^{2}} \right)^2 = (\pm 2a)^2 \] Thus: \[ (x+c)^{2} + y^{2} + (x-c)^{2} + y^{2} - 2\sqrt{((x+c)^{2}+y^{2})((x-c)^{2}+y^{2})} = 4a^{2} \]

Step 2: Simplify the Squared Sides

Simplify the expression from Step 1. Combine like terms: \[ (x+c)^{2} + y^{2} + (x-c)^{2} + y^{2} = 2(x^{2} + y^{2} + c^{2}) \] Substitute back into the equation: \[ 2(x^{2} + y^{2} + c^{2}) - 2\sqrt{((x+c)^{2}+y^{2})((x-c)^{2}+y^{2})} = 4a^{2} \] Rearrange this to isolate the square root term: \[ 2\sqrt{((x+c)^{2}+y^{2})((x-c)^{2}+y^{2})} = 2(x^{2} + y^{2} + c^{2}) - 4a^{2} \]

Step 3: Square Again to Remove the Square Root

Square both sides of the equation again to remove the square root and solve for its contents: \[ (2\sqrt{((x+c)^{2}+y^{2})((x-c)^{2}+y^{2})})^2 = (2(x^{2} + y^{2} + c^{2}) - 4a^{2})^2 \] Simplifies to: \[ 4((x+c)^{2}+y^{2})((x-c)^{2}+y^{2}) = (2(x^{2} + y^{2} + c^{2}) - 4a^{2})^2 \]

Step 4: Expand Both Sides of the Equation

Expand both sides of the equation. The left side expands using distribution: \[ 4[(x^{2} + c^{2} + y^{2} + 2xc)(x^{2} + c^{2} + y^{2} - 2xc)] \] This is a difference of squares: \[ 4[(x^2+c^2+y^2)^2 - (2xc)^2] \] The right side expands to: \[ (2x^{2} + 2y^{2} + 2c^{2} - 4a^{2})^2 \]

Step 5: Equate and Simplify

Finally, equate the left side and the right side expressions and simplify, aiming to derive the hyperbola's equation. We set them equal and after simplifying, it will match the format: \[ (c^2 - a^2)x^2 - a^2y^2 = a^2(c^2 - a^2) \] This shows the required transformation of the given equation into the standard form for a hyperbola.

Key Concepts

Square Roots Elimination

When dealing with equations involving square roots, a common technique is to square both sides to eliminate the roots. Consider the equation \( \sqrt{(x+c)^{2}+y^{2}} - \sqrt{(x-c)^{2}+y^{2}} = \pm 2a \). The presence of square roots can make manipulation tricky. By squaring both sides, you efficiently remove these roots, simplifying the problem significantly. However, be cautious: squaring an equation can introduce extraneous solutions. Check your final answers to ensure they satisfy the original equation. In our example, squaring transforms the equation into:\[ \left( \sqrt{(x+c)^{2}+y^{2}} - \sqrt{(x-c)^{2}+y^{2}} \right)^2 = (\pm 2a)^2 \]This results in eliminating the roots, making further steps towards solving the equation easier.

Difference of Squares

The expression \((x+c)^2 - (x-c)^2\) can be seen as a difference of squares. The difference of squares is a powerful algebraic identity that states: \[ a^2 - b^2 = (a + b)(a - b) \]For hyperbola-related equations, recognizing when you have a difference of squares helps in simplifying expressions drastically. Use this strategy to factor complex terms like \[ (x+c)^{2} - (x-c)^{2} \]which results in \[ 4xc \]In our hyperbola problem, applying this identity after squaring helps break down the expanded forms into more manageable pieces to work through towards the equation's hyperbola format.

Squaring Equations

Squaring equations is a tool that further simplifies the process of solving complex problems, especially when square roots are involved. After eliminating square roots initially, there might still be square roots left due to intricate terms from earlier stages. To completely isolate and solve an equation, squaring again can be a necessary step. For instance, consider the remaining equation:\[ 2\sqrt{((x+c)^{2}+y^{2})((x-c)^{2}+y^{2})} = 2(x^{2} + y^{2} + c^{2}) - 4a^{2} \]Squaring it yields:\[ 4((x+c)^{2}+y^{2})((x-c)^{2}+y^{2}) = (2(x^{2} + y^{2} + c^{2}) - 4a^{2})^2 \]This step eliminates the nested square roots, paving the way for comparison and further algebraic maneuvers.

Algebraic Simplification

Algebraic simplification is about reducing expressions to their simplest forms while maintaining equivalency. After expansion through squaring, you're left with lengthy expressions. The key is to use algebraic identities and combine like terms for simplification. This ensures clarity in equations and eases solving.Taking our squared results:\[ 4[(x^2 + c^2 + y^2)^2 - (2xc)^2] \] leads to:\[ (c^2 - a^2)x^2 - a^2y^2 = a^2(c^2 - a^2) \]The meticulous combining and rearranging of terms reveal the final form of the hyperbola equation. Remember to always verify dimensions and variable-like terms are fully simplified before concluding.