Metric Problems in the Plane in an Arbitrary Affine Coordinate System

Find the tangent of the angle $\alpha$ from the axis $O x$ to the line $y = k x+b$, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$.

Find the tangent of the angle $\alpha$ from the axis $O x$ to the line $y = k x+b$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Find the tangent of the angle $\varphi$ from the line $y = k_{1} x+b_{1}$ to the line $y = k_{2} x+b_{2}$, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$.

Find the tangent of the angle $\varphi$ from the line $y = k_{1} x+b_{1}$ to the line $y = k_{2} x+b_{2}$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Find the necessary and sufficient condition for perpendicularity of the lines $A_{1} x+B_{1} y+C_{1} = 0$ and $A_{2} x+B_{2} y+C_{2} = 0$, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$.

Find the necessary and sufficient condition for perpendicularity of two lines $y = k_{1} x+b_{1}$ and $y = k_{2} x+b_{2}$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1$, $\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Write the equation of the perpendicular dropped from the point $\left(x_{0}, y_{0}\right)$ onto the line $A x+B y+C = 0$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$, set up the equations of the family of lines:

Find the distance $d$ from the point $(x_{0}, y_{0})$ to the line $A x+B y + C = 0$, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$.

Find the distance $d$ from the point $(x_{0}, y_{0})$ to the line $A x+B y + C = 0$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Find the distance $d$ from the point $(2,1)$ to the line $10 x+56 y-37 = 0$, if $g_{11} = 4, g_{12} = 8, g_{22} = 25$.

Set up the equations of the bisectors of the angles between the coordinate axes, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$. Consider the special case $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.