Find so that the distance between (−2,3) and (,1) is √13

We know that the length of the line segment connecting any two points represents the distance between them. There is just one line that connects the two points. Therefore, by measuring the length of the line segment that connects the two points, the distance between them can be determined. If (a, b) and (c, d) be two points, then the distance between them is [tex]\sqrt[]{(b - a)^{2} +(d- c)^{2} }[/tex].

Here, one point is (-2, 3).

Let the other point be (x, 1).

Given that the distance is √13.

Now, [tex]\sqrt[]{(x - (-2))^{2} +(1 - 3)^{2} } = \sqrt{13}[/tex]

i.e. [tex]\sqrt[]{(x + 2)^{2} +( - 2)^{2} } =\sqrt{13}[/tex]

i.e. [tex]\sqrt[]{x^{2}+4x +4 +4 }=\sqrt{13}[/tex]

i.e. [tex]x^{2}+4x +8 =13[/tex]

i.e. [tex]x^{2}+4x + 8- 13=0[/tex]

i.e.[tex]x^{2}+4x -5=0[/tex]

i.e. [tex]x^{2} +5x - x -5=0[/tex]

i.e. [tex]x(x+5)-1(x+5)=0[/tex]

i.e. [tex](x+5)(x-1)=0[/tex]

i.e. [tex]x=-5,1[/tex]

So, the point is either (-5, 1) or (1, 1).

Therefore, the required point is either (-5, 1) or (1, 1).

i.e. the distance between (-2, 3) and (-5, 1) is √13.

or, the distance between (-2, 3) and (1, 1) is √13.

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