Solve the problem.
-If $\mathrm { P } = ( - 6 , - 9 )$ and $\mathrm { Q } = ( \mathrm { x } , - 105 )$, find all numbers $x$ such that the vector represented by $\overrightarrow { \mathrm { PQ } }$ has length $- 120$
A) $\{ 66 , - 78 \}$
B) $\{ 78 , - 78 \}$
C) $\{ 66 , - 84 \}$
D) $\{ 78,66 \}$ .

Question

Quizplus · Accepted Answer

The length of the vector $\overrightarrow{\mathrm{PQ}}$ is given by the distance formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Here, $x_1 = -6$, $y_1 = -9$, $x_2 = x$, and $y_2 = -105$. The length is given as $-120$, but since length cannot be negative, we assume a mistake in the problem statement and use $120$ instead. Plugging in the values, we get:$$120 = \sqrt{(x - (-6))^2 + (-105 - (-9))^2}$$$$120 = \sqrt{(x + 6)^2 + (-96)^2}$$$$120^2 = (x + 6)^2 + 96^2$$$$14400 = (x + 6)^2 + 9216$$$$5184 = (x + 6)^2$$Taking the square root of both sides:$$72 = x + 6 \quad \text{or} \quad -72 = x + 6$$Solving for $x$ gives:$$x = 66 \quad \text{or} \quad x = -78$$Therefore, the correct set of numbers $x$ is $\{66, -78\}$.

Solve the Problem P=(−6,−9)\mathrm { P } = ( - 6 , - 9 )P=(−6,−9)

Solve the Problem $\mathrm { P } = ( - 6 , - 9 )$