Question

Lingkaran yang menyinggung garis x + y = 3 di titik (2,1) dan melalui titik (6,3) mempunyai jari-jari ....

Answer 1

M gris x+y = 3 --> m = -1
misal pusat ling (a,b), jari jari r
garis garis tegak lurus di (2.1) dan (a,b) =m2 = 1

(b-1)/(a-2) = 1
a-2 = b - 1
a - b = 1 .......(1)

pers ling pusat (a,b), r
(x-a)² + (y-b)² - r²
melalui (2,1) --> (2-a)² +(1-b)² = r²
melalui (6,3) --> (6-a)² +(3-b)² = r²

(6-a)² + (3-b)² = (2-a)² +(1-b)²
36-12a+a²+9-6b+b² = 4-4a+a²+1-2b+b²
45-12a-6b = 5-4a-2b
-12a+4a -6b+2b = 5-45
-8a - 4b = - 40 ...bagi (-4)
2a +b = 10 ....(2)

eliminasi (1) dan (2)
a - b = 1
2a+b = 10
----------------(+)
3a = 11
a = 11/3

a-b = 1
b = a - 1 = 11/3 - 1
b = 8/3

lingk melalui (2,1) --> (2-a)² +(1-b)² = r²
(2- 11/3)² + (1- 8/3)² = r²
r² = (-5/3)²+(-5/3)²
r² = 25/9 + 25/9 = 50/9
r = ¹/₃ √50 = ¹/₃ (5√2)
r = ⁵/₃ √2