show that (1,2), (3,-4), (5,-6), (19,8) are concyclic and also find equation of the circle
Answers
YOUR ANSWER IS ☞
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let's equation of the circle =>
=> [ x²+y²+2gx+2fy+c = 0 ]
Now,
put the point in equation of circle
_______
1st => (1,2)
=> 1²+2²+2g(1)+2f(2)+c = 0
=> 5+2g+4f+c = 0
=> 2g+4f+5+c = 0 ________eq(1)
_________
2nd => (3,-4)
=> 3²+(-4)²+2g(3)+2f(-4)+c = 0
=> 9+16+6g-8f+c = 0
=> 6g-8f+25+c = 0 ________eq(2)
_________
3rd => (5,-6)
=> 5²+(-6)²+2g(5)+2f(-6)+c = 0
=> 25+36+10g-12f+c = 0
=> 10g-12f+61+c = 0. ________eq(3)
Now,
_______
by, 2×(eq(1)) + eq(2)
we get..
=> 10g+35+3c = 0 _______eq(4)
_________
by, 3×eq(1) + eq(3)
we get
=> 16g+76+4c = 0 _______eq(5)
_________
by, 3×(eq(5)) - 4×eq(4)
we get
=> 8g + 88 = 0
=> g = -11 _______eq(6)
plug the value of "g" in eq(4)
we get
=> 10(-11)+35+2c = 0
=> -110+35+3c = 0
=> 3c = 75
=> c = 25 __________eq(7)
_________
for "f"
plug the value of "g" and "f" in eq(1)
we get
=> 2(-11)+4f+5+25 = 0
=> 4f+8 = 0
=> f = -2 _____eq(8)
then
_________
equation of circle is
x²+y²-22x-4y+25 = 0
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hope it help you ☺
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DEVIL_KING ▄︻̷̿┻̿═━一
(1,2), (3,-4), (5,-6), (19,8) are concyclic and equation of circle is x²+y²-22x-4y+25=0
Given,
(1,2), (3,-4), (5,-6), (19,8)
To find,
Show that (1,2), (3,-4), (5,-6), (19,8) are concyclic and also find equation of the circle
Solution,
let the equation of the circle be x²+y²+2gx+2fy+c = 0
Now,
put the point in equation of circle
1st => (1,2)
=> 1²+2²+2g(1)+2f(2)+c = 0
=> 5+2g+4f+c = 0
=> 2g+4f+5+c = 0 -(1)
2nd => (3,-4)
=> 3²+(-4)²+2g(3)+2f(-4)+c = 0
=> 9+16+6g-8f+c = 0
=> 6g-8f+25+c = 0 -(2)
3rd => (5,-6)
=> 5²+(-6)²+2g(5)+2f(-6)+c = 0
=> 25+36+10g-12f+c = 0
=> 10g-12f+61+c = 0 -(3)
Now,
by, 2×(1) + (2), we get
=> 10g+35+3c = 0 -(4)
by, 3x(1) + (3), we get
=> 16g+76+4c = 0 -(5)
by, 3x(5) - 4×(4),we get
=> 8g + 88 = 0
=> g = -11 -(6)
putting the value of "g" in (4), we get
=> 10(-11)+35+2c = 0
=> -110+35+3c = 0
=> 3c = 75
=> c = 25 -(7)
for "f"
putting the value of "g" and "f" in (1), we get
=> 2(-11)+4f+5+25 = 0
=> 4f+8 = 0
=> f = -2 -(8)
then
equation of circle is x²+y²-22x-4y+25 = 0
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