If the mid-point of the line joining $(3, 4)$ and $(k, 7)$ is $(x, y)$ and $2x + 2y + 1 = 0$ find the value of $k$.
Given:
The mid-point of the line joining $(3, 4)$ and $(k, 7)$ is $(x, y)$ and $2x + 2y + 1 = 0$.
To do:
We have to find the value of $k$.
Solution:
Mid-point of the line joining the points $(3, 4)$ and $(k, 7)$ is $(x, y)$.
Therefore,
Using mid-point formula, we get,
\( x=\frac{3+k}{2} \) and \( y=\frac{4+7}{2} \)
\( \Rightarrow y=\frac{11}{2} \)
\( 2 x+2 y+1=0 \) (Given)
\( \Rightarrow 2\left(\frac{3+k}{2}\right)+2 \times \frac{11}{2}+1=0 \)
\( \Rightarrow 3+k+11+1=0 \)
\( \Rightarrow k+15=0 \)
\( \Rightarrow k=-15 \)
The value of $k$ is \( -15 \).
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