Question

One diagonal of a square is along the line \[8x - 15y = 0\] and one of its vertexes is \[(1,2)\] then the equation of the sides of the square passing through this vertex, are
A. \[23x + 7y = 9,7x + 23y = 53\]
B. \[23x - 7y + 9 = 0,7x + 23y + 53 = 0\]
C. \[23x - 7y - 9 = 0,7x + 23y - 53 = 0\]
D. None of these

Answer 1

Hint: In this case, we have a line \[8x - 15y = 0\] and a vertex \[(1,2)\] and we are asked to find the equation of the sides of the square passing through this vertex. For this we have to find the equation of sides \[{\rm{BD}}\] with \[{\rm{DC}}\]and proceed with the calculation to find the desired equation.

Formula Used: To describe the point slope form of a line equation, we use the formula
\[{y_1} = m\left( {x - {x_1}} \right)\]

Complete step by step solution: We have been given the data that,
One diagonal of a square is along the line \[8x - 15y = 0\] and one of its vertexes is \[(1,2)\]
Let us assume the square is \[{\rm{ABCD}}\] and vertex \[{\rm{C}}\] is \[(1,2)\]
We have the slope of \[{\rm{BD}}\] is \[\dfrac{8}{{15}}\]
And the angle made by \[{\rm{BD}}\] with \[{\rm{DC}}\] and \[{\rm{DC}}\] is $45^\circ$
Now let us consider the slope of \[{\rm{DC}}\] be \[{\rm{m}}\]
Now, it becomes
\[\tan {45^\circ } = \pm \dfrac{{{\rm{m}} - \dfrac{8}{{15}}}}{{1 + \dfrac{8}{{15}}\;{\rm{m}}}}\]
Now, we have to solve the fraction to simpler term, we have
\[ \Rightarrow (15 + 8\;{\rm{m}}) = \pm (15\;{\rm{m}} - 8)\]
On solving for the value of \[{\rm{m}}\] we get
\[ \Rightarrow {\rm{m}} = \dfrac{{23}}{7}{\rm{ and }} - \dfrac{7}{{23}}\]
Now, we obtained the equations of \[{\rm{DC}}\] and \[{\rm{BC}}\] as
\[ \Rightarrow {\rm{y}} - 2 = \dfrac{{23}}{7}({\rm{x}} - 1)\]
Now, we have to multiply the denominator with the term on the left side and multiply the numerator with the term on the right side, we get
\[ \Rightarrow 23{\rm{x}} - 7{\rm{y}} - 9 = 0\]
\[{\rm{y}} - 2 = - \dfrac{7}{{23}}({\rm{x}} - 1)\]
\[ \Rightarrow 7{\rm{x}} + 23{\rm{y}} - 53 = 0\]
Therefore, the equation of the sides of the square passing through this vertex, are \[23x - 7y - 9 = 0,7x + 23y - 53 = 0\]

Option ‘C’ is correct

Note: Students are likely to make mistakes in these types of problems because it involves trigonometry expressions and more calculations. On should know the formula to proceed with the answer and also should remember all the formulas to solve these types of problems.