Question

If $(1,\frac{P}{3})$ is a mid-point of the line segment joining $(2,0)$ and $(0,\frac{2}{9})$ then show that the line $5x+3y+2$ passes through the point $(-1,3p).$

Answer 1

Let $A(2,0),B(0,\frac{2}{9})$

and let mid-point be $C(1,\frac{p}{3})$

Finding coordinates of $C$

$(1,\frac{p}{3})=(\frac{2+0}{2},\frac{0+\frac{2}{9}}{2})$

$\Rightarrow (1,\frac{p}{3})=(1,\frac{1}{9})$

Comparing y-coordinate

$\frac{p}{3}=\frac{1}{9}$

$\therefore p=\frac{1}{3}$

We need to show that line $5x+3y+2=0$ passes through point $(-1,3p)$

Point $=(-1,3p)=(-1,3\times \frac{1}{3})=(-1,1)$

Putting $(-1,1)$ in line

$5x+3y+2=0$

$\Rightarrow 5(-1)+3\left(1\right)+2=0$

$\Rightarrow -5+3+2=0$

$\Rightarrow -5+5=0$

$\Rightarrow 0=0$

which is true.

Hence, $(-1,3p)$ passes through $5x+3y+2=0$