If point A(x, y) is equidistant from B(3,6) and C(−3,4), find a direct linear relation connecting x and y.
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Step-by-Step Solution
Step 1: Understand the Equidistant Condition
The problem states that point A(x, y) is equidistant from point B(3,6) and point C(−3,4). This means the distance from A to B is equal to the distance from A to C.
Step 2: Apply the Distance Formula
To find the distances AB and AC, we use the distance formula. The distance between two points (x1,y1) and (x2,y2) is given by (x2−x1)2+(y2−y1)2.
Step 3: Set up the Equation
Now, we substitute the coordinates of points A, B, and C into the distance formula. For AB, we use A(x, y) and B(3,6). For AC, we use A(x, y) and C(−3,4). Since AB=AC, we set their distance formulas equal to each other.
Step 4: Square Both Sides and Expand
To eliminate the square roots, we square both sides of the equation. Then, we expand the squared terms using the algebraic identity (a−b)2=a2−2ab+b2 and (a+b)2=a2+2ab+b2.
Step 5: Simplify and Solve for the Relation
We cancel out the x2 and y2 terms from both sides. Then, we gather all terms on one side of the equation and simplify to find the linear relation between x and y. We can divide the entire equation by -4 to get a simpler form.