Question

If the distance between the point of P(2k-1, 2) and Q(k, 2-k) is 25, find the value of k

Answer 1

We use distance formula to solve such type of questions.

The distance between two points whose coordinates are (x1, y1) and (x2, y2) is given by the following formula-

$\underline{ \boxed{ \sf Distance = \sqrt{(x2 - x1)^{2} + (y2 - y1)^{2} } }} \bigstar$

Let us assume the given coordinates as below-

• (x1, y1) = (2k-1 , 2)
• (x2, y2) = (k, 2-k)

By making the use of given formula, we have-

${: \implies\sf Distance = \sqrt{(x2 - x1)^{2} + (y2 - y1)^{2} }}$

${: \implies\sf25= \sqrt{(k - 2k + 1)^{2} + (2 - k - 2)^{2} }}$

${: \implies\sf25= \sqrt{(1 - k)^{2} + ( - k)^{2} }}$

${: \implies\sf25= \sqrt{1 + k^{2} - 2k + k^{2} }}$

${: \implies\sf25= \sqrt{2 k^{2} - 2k + 1}}$

Squaring both sides

${: \implies\sf 625= 2 k^{2} - 2k + 1}$

${: \implies\sf0 = 2 k^{2} - 2k - 624}$

${: \implies\sf0 = k^{2} - k - 312}$

Now, by using quadratic formula we can find the value of this quadratic equation in k.

$: \implies \sf k = \dfrac{ 1 \pm\sqrt{1249} }{2}$

This is your answer.

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