UPSC NDA 2 2021 Mathematics Solved Paper

(a) Slope of line AB ⇒‌

−‌ coefficient of ‌x

‌ coefficient of ‌y

=‌

−1

0

=∞
Slope of line BC=−‌

0

1

=0
∴ Angle between AB and BC
‌‌=|‌

∞−(0)

1+∞⋅(0)

|
⇒‌‌tan‌θ‌‌=∞‌‌|θ=‌

π

2

|

∵△ABC is right angled triangle.
∴ Circumcentre will lie on Hypotenuse
AC i.e. x+2y−4=0 at mid point.
Equation of AB:x−2=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
Equation of AC:x+2y−4=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(ii)
Equation of BC:y+1=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(iii) .
On solving Eqs. (i) and (ii)
‌x=2,y=1
∴‌‌A=(2,1)
‌ On solving Eqs. (ii) and (iii) ‌
‌y=−1,x=6
∴‌‌C‌=(6,−1)
∴‌ Circumcentre will be mid-point of ‌

AC=(‌

2+6

2

,‌

1−1

2

)=(4,0)