Find a point on the x-axis, which is equidistant from the point (… - Vidyadip

Question

Find a point on the x-axis, which is equidistant from the point (7, 6) and (3, 4).

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Answer

It is given that Clie on the x-axis. Let coordinates of C be (x, 0).
Now, C is equidistant from the point A(7, 6) and B(3, 4).
$\therefore \text{ AB}=\text{BC}\ [\text{given}]$
$\Rightarrow \text{AC}^2=\text{BC}^2$
$\Rightarrow \bigg[\sqrt{(\text{x}-7)^2+(0-6)^2}\bigg]= \bigg[\sqrt{(\text{x}-3)^2+(0-4)^2}\bigg]$
$\Rightarrow (\text{x}-7)^2+(-6)^2=(\text{x}-3)^2+(-4)^2$
$\Rightarrow \text{x}^2+49-14\text{x}+36=\text{x}^2+9-6\text{x}+16$
$\Rightarrow 49+36-36-16-9=\text{x}^2-\text{x}^2-6\text{x}+14\text{x}$
$\Rightarrow 85-25=8\text{x}$
$\Rightarrow 60=8\text{x}$
$\Rightarrow 8\text{x}=60$
$\Rightarrow \text{x}=\frac{60}{8}=\frac{15}{2}$
Hence, coordinates of c are $\bigg(\frac{15}{2},0\bigg).$

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