Question
Find the point on x-axis which is equidistant from the points A(3, 2, 2) and B(5, 5, 4).
✓
Answer
We know that the y and z coordinates of the point on the x-axis are 0.
So, let the required point be C(x, y, z)
Now, CA = CB
$\sqrt{(3-\text{x})^2+(2-0)^2+(2-0)^2}=\sqrt{(5-\text{x})^2+(5-0)^2+(4-0)^2}$
$\Rightarrow9-6\text{x}+\text{x}^2+4+4=25-10\text{x}+\text{x}^2+25+16$
$\Rightarrow17-6\text{x}+\text{x}^2=66-10\text{x}+\text{x}^2$
$\Rightarrow4\text{x}=49$
$\Rightarrow\text{x}=\frac{49}{4}$
Hence, the required Point is $\Big(\frac{49}{4},\ 0,\ 0\Big)$
So, let the required point be C(x, y, z)
Now, CA = CB
$\sqrt{(3-\text{x})^2+(2-0)^2+(2-0)^2}=\sqrt{(5-\text{x})^2+(5-0)^2+(4-0)^2}$
$\Rightarrow9-6\text{x}+\text{x}^2+4+4=25-10\text{x}+\text{x}^2+25+16$
$\Rightarrow17-6\text{x}+\text{x}^2=66-10\text{x}+\text{x}^2$
$\Rightarrow4\text{x}=49$
$\Rightarrow\text{x}=\frac{49}{4}$
Hence, the required Point is $\Big(\frac{49}{4},\ 0,\ 0\Big)$
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