Question

a(x,3) and b(4,7) are the two endpoints of a diameter of a circle with central O(3,5).find the value of x​

Answer 1

☯︎ Given that,

• $\sf \bold{A(x , 3)} \: and \: \bold{B(4 , 7)} \: are \: the \: two \: endpoints \: of \: a \: diameter \: of \: a \: circle \: with \: centre \: \bold{O(3 , 5)}$

___________________________

AO and BO are radius of the circle. So, they're equal.

$\bold{\underline{According \: to \: the \: question}}$

• Point O is equidistant from A and B.

$\rm\implies \: AO \: = BO$

$\sf\underline{\: Using \: distance \: formula \: :}$

$\bigstar\boxed{\boxed{\sf\purple{ab \: = \sqrt{x2 - x1) {}^{2} + (y2 - y1) {}^{2} }}}}$

Now,

$\rm\implies \: (3 - x) {}^{2} + (5 - 3) {}^{2} = (3 - 4) {}^{2} + (5 - 7) {}^{2}$

$\rm\implies \: (3) {}^{2} + (x) {}^{2} - 6x + (2) {}^{2} = ( - 1) {}^{2} + ( - 2) {}^{2}$

$\rm\implies \: 9 + x {}^{2} - 6x + 4 = 1 + 4$

$\rm\implies \: 9 + 4 + x {}^{2} - 6x = 5$

$\rm\implies \: 13 + {x}^{2} - 6x = 5$

$\rm\implies \: {x}^{2} - 6x + 8 = 0$

$\rm\implies \: x {}^{2} - 4x - 2x + 8 = 0$

$\rm\implies \: x(x - 4) - 2(x - 4) = 0$

$\rm\implies \: (x - 2)(x - 4) = 0$

$\rm\implies \: x = \boxed{\boxed{\sf\purple{2,4}}}\bigstar$

$\sf\therefore\underline{Hence, \: the \: value \: of \: x \: is \: \bold{2} \: or \: \bold{4}}$

Answer 2

Answer:2

Step-by-step explanation: