Write the coordinates of a point on x-axis which is equidistant f… - Vidyadip

Question

Write the coordinates of a point on x-axis which is equidistant from the points $(-3, 4)$ and $(2, 5).$

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Answer

The point is on $x$-axis.
Its ordinates of the point $P$ is $(x, 0)$.
P is equidistant from $A (-3,4)$ and $B (2,5)$.
Now, $\text{PA}=\sqrt{(\text{x}_2-\text{x}_1)^2+(\text{y}_2-\text{y}_1)^2}$
$=\sqrt{(\text{x}+3)^2+(0-4)^2}=\sqrt{(\text{x}+3)^2+16}$
and $PA^2 = (x + 3)^2 + 16$
Similarly $\text{PB}^2=\big[\sqrt{(\text{x}-2)^2+(0-5)^2}\big]^2$
$= (x - 2)^2 + 25$
$\because$ $PA = PB$
$\Rightarrow PA^2 = PB^2$
$\therefore$ $(x + 3)^2 + 16 = (x - 2)^2 + 25$
$x^2 + 6x + 9 + 16 = x^2 - 4x + 4 + 25$
$\Rightarrow x^2 + 6x - x^2 + 4x = 25 + 4 - 9 - 16$
$\Rightarrow 10x = 4$
$\Rightarrow\ \text{x}=\frac{4}{10}=\frac{2}{5}$
$\therefore$ Co-ordinates of points P will be $\Big(\frac{2}{5},0\Big).$

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