If the points (3, 4) and (X, 1) are equidistant from the origin then the value of x is.

Concept -

For two points to be equidistant from the origin, their distances from the origin must be the same.

The distance between any point (x₁, x₂) and the origin (�1,�1)Unknown node type: span(�1,�1)Unknown node type: span(�1,�1)Unknown node type: span (0,0)(0,0)(0,0) (0,0) is given by the distance formula:

Distance=(�1−0)2+(�1−0)2−−−−−−−−−−−−−−−−−−√=�21+�21−−−−−−−√Distance=(�1−0)2+(�1−0)2−−−−−−−−−−−−−−−−−−√=�21+�21−−−−−−−√Distance=(�1−0)2+(�1−0)2=�12+�12 Distance<mo>=<msqrt><mrow><mo stretchy="false">(<msub><mi>�<mn>1<mo>−<mn>0<msup><mo stretchy="false">)<mn>2<mo>+<mo stretchy="false">(<msub><mi>�<mn>1<mo>−<mn>0<msup><mo stretchy="false">)<mn>2<mo>=<msqrt><mrow><msubsup><mi>�<mn>1<mn>2<mo>+<msubsup><mi>�<mn>1<mn>2</mn></mn></mi></msubsup></mo></mn></mn></mi></msubsup></mrow></msqrt></mo></mn></mo></msup></mn></mo></mn></mi></msub></mo></mo></mn></mo></msup></mn></mo></mn></mi></msub></mo></mrow></msqrt></mo> Distance<mo>=<msqrt><mrow><mo stretchy="false">(<msub><mi>�<mn>1<mo>−<mn>0<msup><mo stretchy="false">)<mn>2<mo>+<mo stretchy="false">(<msub><mi>�<mn>1<mo>−<mn>0<msup><mo stretchy="false">)<mn>2<mo>=<msqrt><mrow><msubsup><mi>�<mn>1<mn>2<mo>+<msubsup><mi>�<mn>1<mn>2</mn></mn></mi></msubsup></mo></mn></mn></mi></msubsup></mrow></msqrt></mo></mn></mo></msup></mn></mo></mn></mi></msub></mo></mo></mn></mo></msup></mn></mo></mn></mi></msub></mo></mrow></msqrt></mo> Distance = 

$\text{Explanation -}$

Given one point is (3, 4), the distance of this point from the origin is:

$\text{Distance = }$

Now, we need to find the value of ��� � � x for the point (�,1)(�,1)(�,1) (x,1) to also have a distance of 5 units from the origin.

Using the distance formula again:

Distance =

Now squaring both side -

⇒ x² + 1 = 25 ⇒ x2 = 24

⇒ x = ± √ 24 = ± 2√ 6

Distance=�2+12−−−−−−−√=Unknown node type: spanDistance=�2+12−−−−−−−√=Unknown node type: spanDistance=�2+12=Unknown node type: span Distance<mo>=<msqrt><mrow><msup><mi>�<mn>2<mo>+<msup><mn>1<mn>2<mo>=<span style="font-family: var(--bs-body-font-family); font-size: var(--bs-body-font-size); font-weight: var(--bs-body-font-weight); text-align: var(--bs-body-text-align);"><strong>5</strong></span></mo></mn></mn></msup></mo></mn></mi></msup></mrow></msqrt></mo> Hence option(2) is correct.