Question

find the values for which distance between points P(3,-5) and Q(k,2) is root 58 units

Answer 1

$\underline{\underline{\Large{\mathfrak{Solution : }}}}$


$\underline{\textsf{Given,}} \\ \\ \sf \implies P(x_1 , y_1) \: = \: P(3,-5) \\ \\ \sf \implies Q(x_2 , y_2) \: = \: Q(k,2) \\ \\ \sf \implies PQ \: = \: \sqrt{58} \: \: units$



$\textsf{Using Distance Formula , } \\ \\ \sf \implies PQ \: = \: \sqrt{(x_1 \: - \: x_2)^2 \: + \: (y_1 \: - \: y_2)^2} \\ \\ \sf \implies \sqrt{58} \: = \: \sqrt{(3 \: - \: k)^2 \: + \: ( - 5 \: - \: 2)^2} \\ \\ \sf \implies {( \sqrt{58} )}^{2} \: = \: {3}^{2} \: + \: {k}^{2} \: - \: 2 \times 3 \times k \: + \: ( - 7)^{2} \\ \\ \sf \implies 58 \: = \: 9 \: + \: {k}^{2} \: - \: 6k \: + \: 49 \\ \\ \sf \implies \cancel{58} \: = \: {k}^{2} \: - \: 6k \: + \: \cancel{ 58} \\ \\ \sf \implies {k}^{2} \: - \: 6k \: = \: 0 \\ \\ \sf \implies k(k \: - \: 6) \: = \: 0$


$\textsf{By Zero Product Rule : } \\ \\ \sf \implies k \: = \: 0 \: \: \quad \implies k \: - \: 6 \: = \: 0 \\ \\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \quad \quad \qquad \implies \: k \: = \: 6 \\ \\ \sf \: \therefore \: k \: = \: 0 \: , \: 6$

Answer 2

$\underline{\underline{\Large{\mathfrak{Solution : }}}} \\\\\\</p><p> </p><p> </p><p> </p><p> </p><p></p><p></p><p>\begin{lgathered}\underline{\textsf{Given,}} \\ \\ \sf \implies P(x_1 , y_1) \: = \: P(3,-5) \\ \\ \sf \implies Q(x_2 , y_2) \: = \: Q(k,2) \\ \\ \sf \implies PQ \: = \: \sqrt{58} \: \: units\end{lgathered} \\\\\\</p><p></p><p> </p><p> </p><p></p><p></p><p></p><p>\begin{lgathered}\textsf{Using Distance Formula , } \\ \\ \sf \implies PQ \: = \: \sqrt{(x_1 \: - \: x_2)^2 \: + \: (y_1 \: - \: y_2)^2} \\ \\ \sf \implies \sqrt{58} \: = \: \sqrt{(3 \: - \: k)^2 \: + \: ( - 5 \: - \: 2)^2} \\ \\ \sf \implies {( \sqrt{58} )}^{2} \: = \: {3}^{2} \: + \: {k}^{2} \: - \: 2 \times 3 \times k \: + \: ( - 7)^{2} \\ \\ \sf \implies 58 \: = \: 9 \: + \: {k}^{2} \: - \: 6k \: + \: 49 \\ \\ \sf \implies \cancel{58} \: = \: {k}^{2} \: - \: 6k \: + \: \cancel{ 58} \\ \\ \sf \implies {k}^{2} \: - \: 6k \: = \: 0 \\ \\ \sf \implies k(k \: - \: 6) \: = \: 0\end{lgathered} \\\\\\</p><p></p><p> </p><p> </p><p></p><p></p><p>\begin{lgathered}\textsf{By Zero Product Rule : } \\ \\ \sf \implies k \: = \: 0 \: \: \quad \implies k \: - \: 6 \: = \: 0 \\ \\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \quad \quad \qquad \implies \: k \: = \: 6 \\ \\ \sf \: \therefore \: k \: = \: 0 \: , \: 6\end{lgathered}$