Question

Find k, given that the point (2,k) is equidistant from (3,7) and (9,1).​

Answer 1

Given : Point (2 , k) is equidistant from (3 , 7) and (9 , 1).

To Find : Value of k.

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Let

$\: \:$

• (2 , k) = P
• (3 , 7) = A
• (9 , 1) = B

$\: \:$

❍ As point P(2 , k) is equidistant from A(3 , 7) and B(9 , 1)

So,

AP = BP

AP² = BP²

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$\:$

As we know that :

$\:$

$\star \: \underline{\boxed{\mathfrak\purple{Distance \: formula = \sqrt{(x_{2} -x_{1} ){}^{2} + (y_{2} - y_{1} ) {}^{2} } }}}$

$\: \:$

$\sf:\implies \: \sqrt{(3 - 2) {}^{2} -(7 - k) {}^{2} } = \sqrt{(9 - 2) {}^{2} - (1 - k) {}^{2} } \\ \\ \\ \sf:\implies \: (3 - 2) {}^{2} - (7 - k) {}^{2} = (9 - 2) {}^{2} - (1 - k) {}^{2} \\ \\ \\ \sf:\implies \: (1) {}^{2} - (49 - 14k + k {}^{2} ) = (7) {}^{2} - (1 - 2k + k {}^{2} ) \\ \\ \\ \sf:\implies \: 1 - 49 + 14k - \cancel{k {}^{2}} = 49 - 1 + 2k - \cancel{k {}^{2} } \\ \\ \\ \sf:\implies \: - 48 + 14k = 48 + 2k \\ \\ \\ \sf:\implies \: 14k - 2k = 48 + 48 \\ \\ \\ \sf:\implies \: 12k = 96 \\ \\ \\ \sf:\implies \: k = \cancel \frac{96}{12} \\ \\ \\ \sf:\implies \: \underline{\boxed{\mathfrak\purple{k = 8}}} \: \bigstar \\ \\ \\ \\ \sf \therefore{ \underline{Hence ,\: the \: value \: of \: k \: is \: \bold{8}.}}$