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If the points $A (4, 3)$ If the pointsand $B (x, 5)$ andare on the circle with center $O (2, 3),$ are on the circle with center then the value of $x 2 + 5$ then the value of is ____.

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Detailed Solution

If the points

A (4, 3)

and

B (x, 5)

are on the circle with center

O (2, 3),

then the value of

x 2 + 5

is 9.
We have been given,
The points

A (4, 3)

and

B (x, 5)

are on the circle with center O(2, 3).
Using distance formula,

D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

We know, the vertices of the triangle are,

A = 4, 3, B = x, 5, O = 2, 3

Calculating the distance of OA,
Substituting

x_{2} = 4, x_{1} = 2, y_{2} = 3, and y_{1} = 3

, in the formula,

OA = \sqrt{{(4 - 2)}^{2} + {(3 - 3)}^{2}}

OA = \sqrt{4}

OA = 2

Calculating distance of OB,
Substituting

x_{2} = x, x_{1} = 2, y_{2} = 5, and y_{1} = 3

, in the formula,

OB = \sqrt{{(x - 2)}^{2} + {(5 - 3)}^{2}}

OB = \sqrt{{(x - 2)}^{2} + 4}

As both are the radius of the same circle, therefore OB = OA.
Now, substituting the values,

(x − 2) 2 + 4 = 2 ⇒ (x − 2) 2 + 4 = 4 ⇒ (x − 2) 2 = 0 ⇒ x − 2 = 0 ⇒ x = 2

Thus, the value of x is 2.
So, substituting the value of x in

x 2 + 5

\Rightarrow

22 + 5

\Rightarrow

4 + 5

\Rightarrow

9
Therefore, the required value is 9.

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