Certainly! We are given an equation and asked to reach a specific form by solving step-by-step. Here’s the detailed solution:
We start with the given proportional relationship:
[tex]\[
\frac{PA}{PB} = \frac{32}{3}
\][/tex]
which implies:
[tex]\[
3PA = 2PB
\][/tex]
We know the distances [tex]\(PA\)[/tex] and [tex]\(PB\)[/tex] represent the Euclidean distance between points [tex]\((x, y)\)[/tex] and points [tex]\(A\)[/tex] [tex]\( (1, 2)\)[/tex] and [tex]\(B\)[/tex] [tex]\((5, 3)\)[/tex] respectively. Therefore, we can rewrite the distances:
[tex]\[
PA = \sqrt{(x-1)^2 + (y-2)^2}
\][/tex]
[tex]\[
PB = \sqrt{(x-5)^2 + (y-3)^2}
\][/tex]
Substituting these into the proportionality condition gives:
[tex]\[
3 \sqrt{(x-1)^2 + (y-2)^2} = 2 \sqrt{(x-5)^2 + (y-3)^2}
\][/tex]
To eliminate the square roots, square both sides of the equation:
[tex]\[
\left( 3 \sqrt{(x-1)^2 + (y-2)^2} \right)^2 = \left( 2 \sqrt{(x-5)^2 + (y-3)^2} \right)^2
\][/tex]
[tex]\[
9 \left( (x-1)^2 + (y-2)^2 \right) = 4 \left( (x-5)^2 + (y-3)^2 \right)
\][/tex]
Expanding both sides of the equation:
[tex]\[
9 \left[ (x-1)^2 + (y-2)^2 \right]
\][/tex]
[tex]\[
= 9 \left( x^2 - 2x \cdot 1 + 1^2 + y^2 - 2y \cdot 2 + 2^2 \right)
\][/tex]
[tex]\[
= 9 \left( x^2 - 2x + 1 + y^2 - 4y + 4 \right)
\][/tex]
[tex]\[
= 9 \left( x^2 + y^2 - 2x - 4y + 5 \right)
\][/tex]
[tex]\[
= 9x^2 + 9y^2 - 18x - 36y + 45
\][/tex]
Now for the right side:
[tex]\[
4 \left[ (x-5)^2 + (y-3)^2 \right]
\][/tex]
[tex]\[
= 4 \left( x^2 - 2x \cdot 5 + 5^2 + y^2 - 2y \cdot 3 + 3^2 \right)
\][/tex]
[tex]\[
= 4 \left( x^2 - 10x + 25 + y^2 - 6y + 9 \right)
\][/tex]
[tex]\[
= 4 \left( x^2 + y^2 - 10x - 6y + 34 \right)
\][/tex]
[tex]\[
= 4x^2 + 4y^2 - 40x - 24y + 136
\][/tex]
Combining both sides, we now have:
[tex]\[
9x^2 + 9y^2 - 18x - 36y + 45 = 4x^2 + 4y^2 - 40x - 24y + 136
\][/tex]
To simplify, bring all terms to one side of the equation:
[tex]\[
9x^2 + 9y^2 - 18x - 36y + 45 - 4x^2 - 4y^2 + 40x + 24y - 136 = 0
\][/tex]
Consolidate like terms:
[tex]\[
(9x^2 - 4x^2) + (9y^2 - 4y^2) + (-18x + 40x) + (-36y + 24y) + (45 - 136) = 0
\][/tex]
[tex]\[
5x^2 + 5y^2 + 22x - 12y - 91 = 0
\][/tex]
Therefore, the final simplified form of the given equation is:
[tex]\[
5x^2 + 5y^2 + 22x - 12y - 91 = 0
\][/tex]
This concludes our solution!
