JEE 2023 Mathematics PYQ — Two circles in the first quadrant of radii and touch the coordina… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Two circles in the first quadrant of radii r1 and r2 touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line x+y=2. Then r12+r22−r1r2 is equal to ___.
Choose the correct answer:
A.
7
(Correct Answer)
B.
8
C.
9
D.
5
Correct Answer:
7
Explanation
1. Circle ki Equation:
Kyuki circle coordinate axes (x aur y axis) ko touch karta hai aur first quadrant mein hai, iska center (r,r) hoga aur radius r hogi. Circle ki equation hogi:
(x−r)2+(y−r)2=r2
x2+y2−2rx−2ry+r2=0
2. Line aur Perpendicular Distance:
Line ki equation hai x+y−2=0.
Center (r,r) se line tak ka perpendicular distance (d) nikalne ka formula:
d=12+12∣r+r−2∣=2∣2r−2∣=2∣r−1∣
3. Intercept Formula:
Intercept ki lambai L=2 di gayi hai. Hum jante hain ki:
(Radius)2=d2+(2L)2
r2=(2(r−1))2+(1)2
r2=2(r2−2r+1)+1
r2=2r2−4r+2+1
r2−4r+3=0
4. Roots aur Final Value:
Yeh quadratic equation r ki do values degi, jo r1 aur r2 hain. Equation r2−4r+3=0 se hum keh sakte hain:
Roots ka sum: r1+r2=4
Roots ka product: r1r2=3
Ab humein r12+r22−r1r2 nikalna hai:
r12+r22−r1r2=(r1+r2)2−2r1r2−r1r2
r12+r22−r1r2=(r1+r2)2−3r1r2
Values rakhne par:
(4)2−3(3)=16−9=7
Final Answer
The value of r12+r22−r1r2 is 7.
Explanation
1. Circle ki Equation:
Kyuki circle coordinate axes (x aur y axis) ko touch karta hai aur first quadrant mein hai, iska center (r,r) hoga aur radius r hogi. Circle ki equation hogi:
(x−r)2+(y−r)2=r2
x2+y2−2rx−2ry+r2=0
2. Line aur Perpendicular Distance:
Line ki equation hai x+y−2=0.
Center (r,r) se line tak ka perpendicular distance (d) nikalne ka formula:
d=12+12∣r+r−2∣=2∣2r−2∣=2∣r−1∣
3. Intercept Formula:
Intercept ki lambai L=2 di gayi hai. Hum jante hain ki:
(Radius)2=d2+(2L)2
r2=(2(r−1))2+(1)2
r2=2(r2−2r+1)+1
r2=2r2−4r+2+1
r2−4r+3=0
4. Roots aur Final Value:
Yeh quadratic equation r ki do values degi, jo r1 aur r2 hain. Equation r2−4r+3=0 se hum keh sakte hain: