Position of two points with respect to a line - Practice Questions & MCQ

Position of two points with respect to a line

Two given points $\mathrm{A}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ and $\mathrm{B}\left(\mathrm{x}_2, \mathrm{y}_2\right)$ lies on the same side of a line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ when $\frac{\mathrm{ax}_1+\mathrm{by}_1+\mathrm{c}}{\mathrm{ax}_2+\mathrm{by}_2+\mathrm{c}}>0 \quad \frac{\mathrm{ax}_1+\mathrm{by}_1+\mathrm{c}}{\mathrm{ax}+0}$ and points lie on the opposite side when

Note:
1. The side of the line where the origin lies is known as the origin side.
2. A point $(\mathrm{p}, \mathrm{q})$ will lie on the origin side of the line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ if $\frac{a p+b q+c}{a \cdot 0+b \cdot 0+c}>0$, meaning ap $+\mathrm{bq}+\mathrm{c}$ and c will have the same sign.
3. A point $(\mathrm{p}, \mathrm{q})$ will lie on the non-origin side of the line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$, if $\frac{a p+b q+c}{a \cdot 0+b .0+c}<0$, meaning $\mathrm{ap}+\mathrm{bq}+\mathrm{c}$ and c will have the opposite sign.

Position of a point which lies inside a triangle
Let $P\left(x_1, y_1\right)$ be the point that lies inside the triangle

The equations of sides of a triangle are

$
\begin{aligned}
& \mathrm{AB}: \mathrm{a}_1 \mathrm{x}+\mathrm{b}_1 \mathrm{y}+\mathrm{c}_1=0 \\
& \mathrm{BC}: \mathrm{a}_2 \mathrm{x}+\mathrm{b}_2 \mathrm{y}+\mathrm{c}_2=0 \\
& \mathrm{CA}: \mathrm{a}_3 \mathrm{x}+\mathrm{b}_3 \mathrm{y}+\mathrm{c}_3=0
\end{aligned}
$
First, find the coordinates of vertices of triangle ABC
Let $A=\left(x^{\prime}, y^{\prime}\right), B=\left(x^{\prime \prime}, y^{\prime \prime}\right)$ and $C=\left(x^{\prime \prime \prime}, y^{\prime \prime \prime}\right)$
And if coordinates of vertices of triangle $A B C$ is given then find equation of sides of triangle $A B C$.
If point $P$ lies inside the triangle, then $P$ and $A$ must be same side of $B C, P$ and $B$ must be same side of $A C$ and $P$ and $C$ must be same side of $A B$, then.

$
\begin{aligned}
& \frac{a_2 x_1+b_2 y_1+c_2}{a_2 x^{\prime}+b_2 y^{\prime}+c_2}>0 \\
& \frac{a_3 x_1+b_3 y_1+c_3}{a_3 x^{\prime \prime}+b_3 y^{\prime \prime}+c_3}>0 \\
& \frac{a_1 x_1+b_1 y_1+c_1}{a_1 x^{\prime \prime \prime}+b_1 y^{\prime \prime \prime}+c_3}>0
\end{aligned}
$