UPSC NDA I 2025 Mathematics Paper

Given vertices A(1,2) and D(−2,6), where AD is the altitude from A onto BC in an equilateral triangle ABC .
Compute the slope of AD:
mAD=‌

yD−yA

xD−xA

=‌

6−2

−2−1

=‌

4

−3

=−‌

4

3

.
Because AD⟂BC, the slope of BC,mBC, satisfies
mAD⋅mBC=−1⟹(−‌

4

3

)mBC=−1⟹mBC=‌

−1

−‌ 4 3

=‌

3

4

Equation of the line BC
‌y−6=‌

3

4

(x+2)
‌4(y−6)=3(x+2)⟹4y−24=3x+6
‌4y−24−3x−6=0
‌⇒−3x+4y−30=0⟹3x−4y+30=0