If center of rhombus is $(\pi, e)$. FInd the equation of diagonal

Question

Two sides of rhombus are parallel to $3x+4y+17=0$ and $4x+3y+16=0$. Center of rhombus is $(\pi, e)$, find the equation of its diagonal.

Is data in this question sufficient to find required diagonal?

Answer 1

Let $\epsilon $ be the angle bisector of the $2$ given lines $\epsilon_1, \epsilon_2$ and $M(x_0,y_0)$ a point on it. Then, the distances of $M$ from the $2$ given lines will be equal. Thus, we have: $$ \begin{array}{c} d(M,\epsilon_1) = d(M,\epsilon_2)\\[2ex] \dfrac{|3x_0 + 4y_0 + 17|}{\sqrt{3^2 + 4^2}} = \dfrac{|4x_0+3y_0+16|}{\sqrt{4^2+3^2}}\\[2ex] |3x_0+4y_0+17| = |4x_0+3y_0+16|\\[2ex] \end{array}$$ From the last equation, we get: $$ y_0 = -x_0 -\frac{33}{7} \text { or } y_0 = x_0 -1.$$

Thus, the slope of the diagonal is either $\lambda = -1$ or $\lambda = 1$. The equation of the diagonal is: $$y-e = -1(x-\pi) \text { or } y-e = 1(x-\pi)\\ \boxed{y =-x+\pi +e} \text { or } \boxed{y = x-\pi + e}.$$

Actually, these are the equations of the 2 diagonals.