(1, –9) is a solution of [tex]x^{2} +y^{2} > 49[/tex], [tex]y[/tex] ≤ [tex]-x^{2} -4[/tex].
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size.
Given inequalities
[tex]x^{2} +y^{2} > 49[/tex]..............(1)
[tex]y[/tex] ≤ [tex]-x^{2} -4[/tex]...............(2)
Option (A)
(–8, –1)
Substitute x = -8, y = -1 in equation 1
[tex]x^{2} +y^{2} > 49[/tex]
⇒ [tex](-8)^{2}+(-1)^{2} > 49[/tex]
L.H.S = 65
R.H.S = 49
65 > 49
It is true
Substitute x = -8, y = -1 in equation 2
[tex]y[/tex] ≤ [tex]-x^{2} -4[/tex]
-1 ≤ [tex]-(-8)^{2} -4[/tex]
L.H.S = -1
R.H.S = -68
-1 ≤ -68
It is false.
It is not suitable solution.
Option (B)
(–3, 1)
Substitute x = -3, y = 1 in equation 1
[tex]x^{2} +y^{2} > 49[/tex]
⇒ [tex](-3)^{2}+(1)^{2} > 49[/tex]
L.H.S = 10
R.H.S = 49
10 > 49
It is false
Substitute x = -3, y = 1 in equation 2
[tex]y[/tex] ≤ [tex]-x^{2} -4[/tex]
1 ≤ [tex]-(-3)^{2} -4[/tex]
L.H.S = 1
R.H.S = -13
1 ≤ -13
It is false.
It is not suitable solution.
Option (C)
(1, –9)
Substitute x = 1, y = -9 in equation 1
[tex]x^{2} +y^{2} > 49[/tex]
⇒ [tex]1^{2}+(-9)^{2} > 49[/tex]
L.H.S = 82
R.H.S = 49
82 > 49
It is true
Substitute x = 1, y = -9 in equation 2
[tex]y[/tex] ≤ [tex]-x^{2} -4[/tex]
-9 ≤ [tex](-1)^{2} -4[/tex]
L.H.S = -9
R.H.S = -5
-9 ≤ -5
It is true.
Hence, (1, –9) is a solution of [tex]x^{2} +y^{2} > 49[/tex], [tex]y[/tex] ≤ [tex]-x^{2} -4[/tex].
Option (D)
(0, –6)
Substitute x = 0, y = -6 in equation 1
[tex]x^{2} +y^{2} > 49[/tex]
⇒ [tex]0^{2}+(-6)^{2} > 49[/tex]
L.H.S = 36
R.H.S = 49
36 > 49
It is false
Substitute x = 0, y = -6 in equation 2
[tex]y[/tex] ≤ [tex]-x^{2} -4[/tex]
-6 ≤ [tex]-(0)^{2} -4[/tex]
L.H.S = -4
R.H.S = -6
-4 ≤ -6
It is false.
It is not suitable solution.
Hence, (1, –9) is a solution of [tex]x^{2} +y^{2} > 49[/tex], [tex]y[/tex] ≤ [tex]-x^{2} -4[/tex].
Option C is correct.
Find out more information about inequalities here
https://brainly.com/question/26852790
#SPJ3
