Answer

Verified

387.6k+ views

**Hint:**Here, we have to find the value of the variable \[x\]. We have \[\left( {x + 2,x - 1} \right)\] as the center of circle and two points on the circumference of the circle are given as \[\left( {2, - 2} \right)\& \left( {8, - 2} \right)\].we know the distance of the center of the circle from any point on the circumference of the circle is equal to radius. Thus, the distance of the point \[\left( {x + 2,x - 1} \right)\] from the two given points \[\left( {2, - 2} \right)\& \left( {8, - 2} \right)\] is equal.

**Complete Complete Step by Step Solution:**

First we will draw the diagram of the circle as per the given information.

Given that, a circle passes through points \[\left( {2, - 2} \right)\& \left( {8, - 2} \right)\].

This means that these points lie on the circumference of the circle.

We know that the distance from the center of the circle to any point on the circumference is equal to each other as it is called as radius.

So the distance of both the points from the center is equal.

Now, we will calculate the distance between the point \[\left( {x + 2,x - 1} \right)\] and \[\left( {2, - 2} \right)\] using the distance formula.

Distance of the point \[\left( {x + 2,x - 1} \right)\]and \[\left( {2, - 2} \right)\] \[ = \sqrt {\left[ {{{\left( {x + 2 - 2} \right)}^2} + {{\left( {x - 1 + 2} \right)}^2}} \right]} \]

On further simplification, we get

\[ \Rightarrow \] Distance of the point \[\left( {x + 2,x - 1} \right)\] and \[\left( {2, - 2} \right)\] \[ = \sqrt {\left[ {{x^2} + {{\left( {x + 1} \right)}^2}} \right]} \]……………….\[\left( 1 \right)\]

Now, we will calculate the distance between the point \[\left( {x + 2,x - 1} \right)\] and \[\left( {8, - 2} \right)\] using the distance formula.

Distance of the point \[\left( {x + 2,x - 1} \right)\]and \[\left( {8, - 2} \right)\] \[ = \sqrt {\left[ {{{\left( {x + 2 - 8} \right)}^2} + {{\left( {x - 1 + 2} \right)}^2}} \right]} \]

On further simplification, we get

\[ \Rightarrow \] Distance of the point \[\left( {x + 2,x - 1} \right)\] and \[\left( {8, - 2} \right)\] \[ = \sqrt {\left[ {{{\left( {x - 6} \right)}^2} + {{\left( {x + 1} \right)}^2}} \right]} \]……………….\[\left( 2 \right)\]

Equating equation 1 and equation 2 as both are the radius of the same circle, we get

\[\sqrt {\left[ {{x^2} + {{\left( {x + 1} \right)}^2}} \right]} = \sqrt {\left[ {{{\left( {x - 6} \right)}^2} + {{\left( {x + 1} \right)}^2}} \right]} \]

Squaring on both sides, we cancel the roots. Hence we get,

\[ \Rightarrow {x^2} + {\left( {x + 1} \right)^2} = {\left( {x - 6} \right)^2} + {\left( {x + 1} \right)^2}\]

On simplifying the terms, we get

\[ \Rightarrow {x^2} = {\left( {x - 6} \right)^2}\]

Applying exponents on the bases, we get

\[ \Rightarrow {x^2} = {x^2} - 36 + 12x\]

Rearranging the terms, we get

\[\begin{array}{l} \Rightarrow 12x = 36\\ \Rightarrow x = 3\end{array}\]

**Hence, the value of x is 3.**

**Note:**

1) The formula which we have used here for finding the distance between the two points is in two dimensional.

2) The distance formula for three dimensional is different.

3) Say we have to find the distance between the points \[({x_1},{y_1},{z_1})\] and \[({x_2},{y_2},{z_2})\]

4) So the distance between these two points is

\[\sqrt {\left[ {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{({z_2} - {z_1})}^2}} \right]} \]

Recently Updated Pages

The principal kept the staff members waiting Change class 8 english CBSE

Perseverance is the key to success Expand the following class 8 english CBSE

Which local body has a Mayor as its head class 8 social science CBSE

Is the word stars countable or uncountable class 8 english CBSE

Is the word scrupulous a positive negative or neutral class 8 english CBSE

Are days of the week proper nouns class 8 english CBSE

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

10 examples of law on inertia in our daily life