Find a relation between x and y such that the point (x, y) is equidistant from the points (7,1) and (3,5).
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Step-by-Step Solution
Step 1: Define the points
Let the point (x, y) be denoted as P. Let the two given points be A(7,1) and B(3,5). We are given that point P is equidistant from A and B, which means the distance PA is equal to the distance PB.
Step 2: Set up the distance equality
Since point P is equidistant from A and B, their distances must be equal. To avoid dealing with square roots, we can square both sides of the equation. This leads to PA2=PB2.
Step 3: Apply the distance formula
We use the distance formula to express PA2 and PB2. The distance formula states that the distance between two points (x1,y1) and (x2,y2) is (x2−x1)2+(y2−y1)2. Squaring this removes the square root, giving us (x2−x1)2+(y2−y1)2.
Step 4: Equate the squared distances
Now we equate the expressions for PA2 and PB2 as established in the previous steps. This equation forms the basis for finding the relation between x and y.
Step 5: Expand the terms
We expand each squared term using the algebraic identity (a−b)2=a2−2ab+b2. This allows us to remove the parentheses and prepare for simplification.
Step 6: Simplify the equation
Combine the constant terms on each side. We have 49+1=50 on the left side and 9+25=34 on the right side. We also observe that x2 and y2 terms are present on both sides, and they will cancel out.
Step 7: Cancel common terms
We cancel the x2 and y2 terms from both sides of the equation, as they are identical. This simplifies the equation significantly, leaving only linear terms in x and y and constant terms.
Step 8: Rearrange and solve for the relation
Now, we gather all the x terms and y terms on one side and the constant terms on the other side. This rearrangement allows us to isolate the variables and simplify the equation into the desired relation between x and y. Finally, divide by 8 to get the simplest form.