Question

(a+b,a-b) find the distance of the point from origin

Answer 1

Answer:

The distance of the given point from the origin is

$\displaystyle{\boxed{\red{\sf\:\sqrt{2\:(\:a^2\:+\:b^2\:)}\:units\:}}}$

Step-by-step-explanation:

Let the given point be A.

And the origin be O.

• O ≡ ( 0, 0 ) ≡ ( x₁, y₁ )
• A ≡ ( a + b, a - b ) ≡ ( x₂, y₂ )

We have to find the distance between these two points.

Now, by distance formula,

$\displaystyle{\boxed{\pink{\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{(\:x_1\:-\:x_2\:)^2\:+\:(\:y_1\:-\:y_2\:)^2}}}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{[\:0\:-\:(\:a\:+\:b\:)\:]^2\:+\:[\:0\:-\:(\:a\:-\:b\:)\:]^2}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{(\:0\:-\:a\:-\:b\:)^2\:+\:(\:0\:-\:a\:+\:b\:)^2}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{(\:-\:a\:-\:b\:)^2\:+\:(\:-\:a\:+\:b\:)^2}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{[\:(\:-\:a\:)^2\:-\:2\:\times\:(\:-\:a\:)\:\times\:b\:+\:b^2\:]\:+\:[\:(\:-\:a\:)^2\:+\:2\:\times\:(\:-\:a\:)\:\times\:b\:+\:b^2}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{(\:a^2\:+\:2ab\:+\:b^2\:)\:+\:(\:a^2\:-\:2ab\:+\:b^2\:)}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{a^2\:+\:\cancel{2ab}\:+\:b^2\:+\:a^2\:-\:\cancel{2ab}\:+\:b^2}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{a^2\:+\:b^2\:+\:a^2\:+\:b^2}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{a^2\:+\:a^2\:+\:b^2\:+\:b^2}}$

$\displaystyle{\implies\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{2a^2\:+\:2b^2}}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:d\:(\:O\:,\:A\:)\:=\:\sqrt{2\:(\:a^2\:+\:b^2\:)}\:}}}}$

∴ The distance of the given point from the origin is

$\displaystyle{\boxed{\red{\sf\:\sqrt{2\:(\:a^2\:+\:b^2\:)}\:units\:}}}$

Answer 2

$\rm \color{orange}{Distance \: \: \: from \: \: Origin \: } \color{silveryblue}= \color{green}\sqrt{2( {a}^{2} + {b}^{2} )}$