Question

M(-4,3) and N(x,6) if the magnitude of |MN| is 3√10.find the value of x

Answer 1

To find the value of \( x \) given the points \( M(-4,3) \) and \( N(x,6) \) where the distance between them \( |MN| \) is \( 3\sqrt{10} \), we use the distance formula:

\[
|MN| = \sqrt{(x - (-4))^2 + (6 - 3)^2}
\]

This simplifies to:

\[
|MN| = \sqrt{(x + 4)^2 + (3)^2}
\]

Now, since \( |MN| = 3\sqrt{10} \), we set up the equation:

\[
\sqrt{(x + 4)^2 + 3^2} = 3\sqrt{10}
\]

Squaring both sides, we eliminate the square root:

\[
(x + 4)^2 + 9 = (3\sqrt{10})^2
\]

Calculating \( (3\sqrt{10})^2 \):

\[
(3\sqrt{10})^2 = 9 \cdot 10 = 90
\]

Now we can rewrite the equation:

\[
(x + 4)^2 + 9 = 90
\]

Subtracting 9 from both sides gives:

\[
(x + 4)^2 = 81
\]

Taking the square root of both sides, we have:

\[
x + 4 = 9 \quad \text{or} \quad x + 4 = -9
\]

This leads to two possibilities for \( x \):

1. \( x + 4 = 9 \)
\[
x = 9 - 4 = 5
\]

2. \( x + 4 = -9 \)
\[
x = -9 - 4 = -13
\]

Thus, the two possible values of \( x \) are:

\[
\boxed{5} \quad \text{or} \quad \boxed{-13}
\]