To find the value of \( x \) for point \( B(x, y) \) such that the distance between points \( A(1, -2) \) and \( B(x, y) \) is 5, we can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, the distance \( d = 5 \), \( A(1, -2) \) has coordinates \( (x_1, y_1) = (1, -2) \), and \( B \) has coordinates \( (x_2, y_2) = (x, y) \).
Substituting into the distance formula:
\[ 5 = \sqrt{(x - 1)^2 + (y + 2)^2} \]
Now squaring both sides to eliminate the square root gives:
\[ 25 = (x - 1)^2 + (y + 2)^2 \]
Now we could express this as:
\[ (x - 1)^2 + (y + 2)^2 = 25 \]
This represents a circle centered at \( (1, -2) \) with a radius of 5. There are infinitely many points \( B(x, y) \) that satisfy this equation. To find specific values for \( x \), we need either the \( y \) coordinate or any additional conditions for \( B \).
If you just want to find values of \( x \) while \( y \) is arbitrary, you can set \( y \) to particular values:
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If \( y = -2 \): \[ (x - 1)^2 + 0 = 25 \implies (x - 1)^2 = 25 \implies x - 1 = 5 \text{ or } x - 1 = -5 \] Thus, \( x = 6 \) or \( x = -4 \).
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If \( y = 3 \): \[ (x - 1)^2 + 5^2 = 25 \implies (x - 1)^2 + 25 = 25 \implies (x - 1)^2 = 0 \implies x - 1 = 0 \] Thus, \( x = 1 \).
Therefore, depending on the value of \( y \), the possible values of \( x \) include \( 6, -4, \) and others based on different choices of \( y \). The general requirement is that \( (x - 1)^2 + (y + 2)^2 = 25 \) constitutes all potential coordinate pairs \( (x, y) \) on that circle.
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