CUET PG 2026 Mathematics PYQ — Which of the following option is false about circles whose equati… | Mathem Solvex | Mathem Solvex
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CUET PG 2026 — Mathematics PYQ
CUET PG | Mathematics | 2026
Which of the following option is false about circles whose equation x2+y2−10x−10y+41=0, x2+y2−22x−10y+137=0
(A) circles have same center (B) circles have no meeting point (C) circles have only one meeting point (D) circles have only two meeting point
Choose the correct answer:
A.
A, B, C
B.
B, C, D
C.
A, B, D
(Correct Answer)
D.
A, C, D
Correct Answer:
A, B, D
Explanation
Solution
To find the relationship between two circles, we first find their centers (h,k) and radii (r) using the general form x2+y2+2gx+2fy+c=0, where Center =(−g,−f) and Radius =g2+f2−c.
1. Analysis of Circle 1 (S1):
Equation: x2+y2−10x−10y+41=0
g=−5,f=−5,c=41
Center C1=(5,5)
Radius r1=(−5)2+(−5)2−41=25+25−41=9=3
2. Analysis of Circle 2 (S2):
Equation: x2+y2−22x−10y+137=0
g=−11,f=−5,c=137
Center C2=(11,5)
Radius r2=(−11)2+(−5)2−137=121+25−137=9=3
3. Comparison:
Centers:C1(5,5) and C2(11,5). They are not the same. So, Statement (A) is False.
Distance between centers (d):
d=(11−5)2+(5−5)2=62+02=6
Sum of Radii (r1+r2):
r1+r2=3+3=6
4. Conclusion on Meeting Points:
Since the distance between centers d is exactly equal to the sum of the radii (d=r1+r2), the two circles touch each other externally at exactly one point.
Statement (B) says "no meeting time" →False
Statement (C) says "only one meeting time" →True
Statement (D) says "only two meeting time" →False
Identifying False Statements:
The false statements are A, B, and D.
Correct Option: (c)
Explanation
Solution
To find the relationship between two circles, we first find their centers (h,k) and radii (r) using the general form x2+y2+2gx+2fy+c=0, where Center =(−g,−f) and Radius =g2+f2−c.
1. Analysis of Circle 1 (S1):
Equation: x2+y2−10x−10y+41=0
g=−5,f=−5,c=41
Center C1=(5,5)
Radius r1=(−5)2+(−5)2−41=25+25−41=9=3
2. Analysis of Circle 2 (S2):
Equation: x2+y2−22x−10y+137=0
g=−11,f=−5,c=137
Center C2=(11,5)
Radius r2=(−11)2+(−5)2−137=121+25−137=9=3
3. Comparison:
Centers:C1(5,5) and C2(11,5). They are not the same. So, Statement (A) is False.
Distance between centers (d):
d=(11−5)2+(5−5)2=62+02=6
Sum of Radii (r1+r2):
r1+r2=3+3=6
4. Conclusion on Meeting Points:
Since the distance between centers d is exactly equal to the sum of the radii (d=r1+r2), the two circles touch each other externally at exactly one point.
