Find \(a\) if the distance between \(A(-3,-14)\) and \(B(a,-5)\) is \(9\).
\(a=-3\)
Step 1: Recall the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 2: Here, the coordinates are:
Step 3: Substitute values into the formula:
\[ 9 = \sqrt{(a - (-3))^2 + (-5 - (-14))^2} \]
Step 4: Simplify the terms:
So:
\[ 9 = \sqrt{(a+3)^2 + (9)^2} \]
Step 5: Square both sides to remove the square root:
\[ 9^2 = (a+3)^2 + 9^2 \]
\[ 81 = (a+3)^2 + 81 \]
Step 6: Subtract 81 from both sides:
\[ 81 - 81 = (a+3)^2 \]
\[ 0 = (a+3)^2 \]
Step 7: Take square root of both sides:
\[ a + 3 = 0 \]
Step 8: Solve for \(a\):
\[ a = -3 \]
Final Answer: \(a = -3\).