Question

The opposite vertices of a square are \[\left( {1,2} \right)\] and \[\left( {3,8} \right)\]. What is the equation of a diagonal of the square passing through the point \[\left( {1,2} \right)\] is
A. \[3x - y - 1 = 0\]
B. \[3y - x - 1 = 0\]
C. \[3x + y + 1 = 0\]
D. None of these

Answer 1

Hint: Two opposite vertices of a square are given. You have to find the equation of a diagonal of the square which passes through one of the given vertices. Use point slope form to find the equation. The slope can be obtained using the formula of finding the slope of a line through two points and a point is given. Using these the equation of the line can be obtained easily.

Formula used
Slope of a line passing through two points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] is \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Equation of a line having slope \[m\] and passing through the point \[\left( {{x_1},{y_1}} \right)\] is \[\dfrac{{y - {y_1}}}{{x - {x_1}}} = m\]

Complete step by step solution
Given that two opposite vertices of a square are \[\left( {1,2} \right)\] and \[\left( {3,8} \right)\].
Let \[A = \left( {1,2} \right),C = \left( {3,8} \right)\]
Then \[AC\] is a diagonal of the square.
Slope of the diagonal \[AC\] is \[{m_1} = \dfrac{{8 - 2}}{{3 - 1}} = \dfrac{6}{2} = 3\]
This diagonal passes through the point \[\left( {1,2} \right)\].
So, the equation of the diagonal is
\[\dfrac{{y - 2}}{{x - 1}} = 3\]
\[ \Rightarrow y - 2 = 3x - 3\]
\[ \Rightarrow 3x - 3 = y - 2\]
\[ \Rightarrow 3x - y - 3 + 2 = 0\]
\[ \Rightarrow 3x - y - 1 = 0\]

Hence option A is correct.

Note: Look at the question carefully. In this question, such an equation of a line is required which passes through a point already given. So, you have to find the slope of the line only.