Question

Find the distances between the following pair of points. $(a{m}_1^2,2a{m}_1)$ and $(a{m}_2^2,2a{m}_2)$.

Answer 1

Distance $d$ between two points $P(a,b)$ and $Q(c,d)$ is given by,

$d=\sqrt{(a-c{)}^2+(b-d{)}^2}$

Given that :

$P(a{m}_1^2,2a{m}_1)$

$Q(a{m}_2^2,2a{m}_2)$

Now apply the above given formula,

$d=\sqrt{(a{m}_1^2-a{m}_2^2{)}^2+(2a{m}_1-2a{m}_2{)}^2}$

$d=a\sqrt{(m_1^2-m_2^2{)}^2+(2m_1-2m_2{)}^2}$

$d=a\sqrt{{\left[(m_1+m_2)(m_1-m_2)\right]}^2+4(m_1-m_2{)}^2}$

Taking $(m_1-m_2{)}^2$ common we get,

$\Rightarrow d=a(m_1-m_2)\sqrt{(m_1+m_2{)}^2+4}$