find the point on the x-axis which is equidistant from point A ( -3,4) and B ( 1,4)​

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1. Basic Concept Used :-

   Distance Formula :-

   This Formula is used to find the distance between two given points.

   Let us assume a line segment joining the points A and B, then distance between A and B is given by

   [tex] \sf \: \:AB = \sqrt{ {(x_2-x_1)}^{2} + {(y_2-y_1)}^{2} }[/tex]

   [tex] \sf \: where \: coordinates \: of \: A \: and \: B \: are \: (x_1,y_1) \: and \: (x_2,y_2)[/tex]

   Let’s solve the problem now!!

   • Let suppose that the point on x – axis be P (x, 0)

   Given coordinates are

   • A (- 3, 4)

   and

   • B (1, 4)

   According to statement,

   • Point P is equidistant from A and B,

   [tex]\bf :\implies\:PA = PB[/tex]

   On squaring both sides, we get

   [tex]\rm :\longmapsto\: {PA}^{2} = {PB}^{2} [/tex]

   [tex]\rm :\longmapsto\: {(x + 3)}^{2} + \cancel{{(y – 4)}^{2}} = {(x – 1)}^{2} + \cancel{(y – 4)}^{2} [/tex]

   [tex]\rm :\longmapsto\: \cancel{x}^{2} + 9 – 6x = \cancel{x}^{2} + 1 + 2x[/tex]

   [tex]\rm :\longmapsto\: – 6x – 2x = 1 – 9[/tex]

   [tex]\rm :\longmapsto\: – 8x = – 8[/tex]

   [tex]\bf\implies \:x = 1[/tex]

   [tex]\bf:\implies\:\: Point \: on \: x – axis \: be \: P \: (1, \: 0)[/tex]

   Additional Information :-

   [tex]\underline{\bigstar\:\textsf{Section Formula\; :}}[/tex]

   Section Formula is used to find the co ordinates of the point C (x, y) which divides the line segment joining the points (A) and (B) internally in the ratio m : n, then

   [tex]{\underline{\boxed{\sf{\quad \Big(x, y \Big) = \Bigg(\dfrac{mx_2 + nx_1}{m + n}, \: \dfrac{my_2 + ny_1}{m + n}\Bigg) \quad}}}}[/tex]

   [tex] \sf \: where \: coordinates \: of \: A \: and \: B \: are \: (x_1,y_1) \: and \: (x_2,y_2)[/tex]

   [tex]\underline{\bigstar\:\textsf{Mid Point Formula\; :}}[/tex]

   Mid Point formula is used to find the Mid points on any line.

   Let us assume a line segment joining the points A and B and let C be the midpoint of AB, then coordinates of C is

   [tex]{\underline{\boxed{\sf{\quad \bigg(\dfrac{x_1 + x_2}{2} \; ,\; \dfrac{y_1 + y_2}{2} \bigg) \quad}}}}[/tex]

   [tex] \sf \: where \: coordinates \: of \: A \: and \: B \: are \: (x_1,y_1) \: and \: (x_2,y_2)[/tex]

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