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in Coordinate Geometry by (28.9k points)
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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.

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Let A(3, 4) and B(k, 7) and midpoint be C(x, y) which lies on the line 2x + 2y +1 = 0

By midpoint formula,

x = \(\frac{x_1 + x_2}2\), y = \(\frac{y_1 + y_2}2\)

For point C(x, y),

x = \(\frac{3 + k}2\), y = \(\frac{4 + 7}2\)....(1)

Here, y = \(\frac{11}2,\)

Hence, substituting value of y in given equation of line,

2x + 2 × \(\frac{11}2 + 1\) = 0

∴ 2x = -12 

∴ x = - 6 

Now substituting value of x in equation(1), we get.

x = \(\frac{3 + k}2\)

- 6 = \(\frac{3 + k}2\)

∴ -12 = 3 + k 

∴ k = -15 

Hence, the value of k is -15.

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