Question

please help me to solve from this step.... coordinate geometry

Answer 1

See, under the root, there are two terms:

$first \: term : \: {a}^{2} ( {y - x})^{2} (y + x)^{2} \\ second \: term : \: 4{a}^{2} ( {y - x})^{2}$

What is common between the two terms?

Common between two terms are a^2 and (y-x)^2

So if you take them common,
in the first term, (y+x)^2 will remain
in the second term, 4 will remain

So after taking common, it will be

$( {a}^{2} ) \times {(y - x)}^{2} \times [(y + x)^{2} + 4]$

So AB will be

$AB = \sqrt{( {a}^{2} ) \times {(y - x)}^{2} \times [(y + x)^{2} + 4]} \\ \\ AB = a \times (y - x) \times \sqrt{(y + x)^{2} + 4}$

$AB = a (y - x) \sqrt{(y + x)^{2} + 4}$