Question

Find the point on the x-axis which is equiditant from 2,-5 and -2, 9​

Answer 1

$\LARGE{ \underline{\underline{ \bf{Required \: answer:}}}}$

We have to find the point on x axis which will be equidistant from the points (2,-5) and (-2,9):

We know that the ordinate of all points on x-axis is 0. Then the coordinates of the point is (x,0)

Distance between point (x,0) and (2,-5) is equals to the distance between point (-2,9) and (x,0)

By using distance formula,

$\cdot{ \boxed{ \sf{d = \sqrt{(x2 - x1) {}^{2} + (y2 - y1) {}^{2} } }}}$

Plugging the given values:

⇒ $\sf{ \sqrt{(x - 2) {}^{2} + (0 + 5) {}^{2} } = \sqrt{( - 2 - x) {}^{2} + {9}^{2} } }$

Squaring both sides,

⇒ $\sf{ (x - 2) {}^{2} + {5}^{2} = (2 + x) {}^{2} + {9}^{2} }$

Calculating further,

⇒ $\sf{ {x}^{2} - 4x + 4 + 25 = {x}^{2} + 4x + 4 + 81}$

⇒ $\sf{ {x}^{2} - 4x + 29 = {x}^{2} + 4x + 85}$

Now shifting the like terms to one side of the eq.

⇒ $\sf{ {x}^{2} - {x}^{2} - 4x - 4x + 29 - 85 = 0}$

⇒ $\sf{ - 8x - 56 = 0}$

⇒ $\sf{ - 8(x + 7) = 0}$

⇒ $\sf{x = - 7}$

Hence,

• The coordinate of the point on x-axis which is equidistant from points (2,-5) and (-2,9) is (-7,0)

Answer 2

$\rule{220}{1}$

${ \huge{ \boxed{ \overline{ \mid{ \sf{ \orange{Answer : }}}}}}}$

$\sf \: by \: using \: distance \: formula,$

$\rightarrow\sf{d = \sqrt{(x2 - x1) {}^{2} + (y2 - y1) ^{2}} }$

$\rightarrow\sf{ \sqrt{(x - 2)^{2} (0 + 5) ^{2} } + = \sqrt{( - 2 - x) ^{2} + {9}^{2} } }$

$\sf {squaring \: the \: both \: sides,}$

$\rightarrow\sf{(x - 2) ^{2} + 5 ^{2} = (2 + x) ^{2} + {9}^{2} }$

$\rightarrow\sf{ { x }^{2} - 4x + 29 = {x}^{2} + 4x + 85}$

$\sf{Take\: the\: like \:terms\: at\: one \:side,}$

$\rightarrow\sf{ { x}^{2} - {x}^{2} - 4x - 4x + 29 - 85 }$

$\sf{simplify\:it,}$

$\rightarrow\sf{ - 8x - 56 = 0}$

$\rightarrow\sf{ - 8(x + 7) = 0}$

$\rightarrow\sf{x = - 7}$

$\sf{hence ,\: the \: the \: point \: on \: x \: axis \: is \: ( - 7 \: ,0)}$

$\sf{The\:points\:are,}$

${ \huge{ \boxed{ \overline{ \mid{ \sf{(2, -5) }}}}}}$

${ \huge{ \boxed{ \overline{ \mid{ \sf{(-2, 9 )}}}}}}$

${ \huge{ \boxed{\boxed{ \overline{ \mid{ \sf{ \orange{(-7, 0 )}}}}}}}}$

$\rule{220}{1}$