The point B is required and it needs to be verified.
The point is [tex]B(7,5)[/tex]
It is verified below.
The points are
[tex]P(1,2)[/tex] and [tex]A(-3,0)[/tex]
[tex]x_1=-3[/tex]
[tex]y_1=0[/tex]
The ratio is assumed to be internal
[tex]m:n=2:3[/tex]
From the partition formula
[tex](1,2)=\left(\dfrac{2\times x_2+3\times -3}{2+3},\dfrac{2\times y_2+3\times 0}{2+3}\right)[/tex]
[tex]1=\dfrac{2\times x_2+3\times -3}{2+3}\\\Rightarrow 5=2x_2-9\\\Rightarrow x_2=\dfrac{5+9}{2}\\\Rightarrow x_2=7[/tex]
[tex]2=\dfrac{2\times y_2+3\times 0}{2+3}\\\Rightarrow \dfrac{2\times 5-0}{2}\\\Rightarrow y_2=5[/tex]
The required point is [tex]B(7,5)[/tex]
The equation of the line is
[tex]y-2=\dfrac{1}{2}(x-1)\\\Rightarrow y=\dfrac{1}{2}x-\dfrac{1}{2}+2\\\Rightarrow y=\dfrac{1}{2}x+\dfrac{3}{2}[/tex]
Verifying
[tex]y=5[/tex]
[tex]5=\dfrac{1}{2}x+\dfrac{3}{2}\\\Rightarrow x=\dfrac{5-\dfrac{3}{2}}{\dfrac{1}{2}}\\\Rightarrow x=7[/tex]
It is verified.
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