NDA 1 2025 Maths, question 55

Topic: Analytical geometry

Q55

$ABC$ is an equilateral triangle and $AD$ is the altitude on $BC$ . If the coordinates of $A$ are $(1,2)$ and that of $D$ are $( - 2,6)$ , then what is the equation of $BC$ ?

(a) $3x + 4y - 18 = 0$
(b) $4x + 3y - 1 = 0$
(c) $4x - 3y + 26 = 0$
(d) $3x - 4y + 30 = 0$ Correct answer

Solution.

Answer: (d) $3x - 4y + 30 = 0$

An altitude is perpendicular to the opposite side.
The vector $\overrightarrow{DA} = (3, - 4)$ is therefore a normal to $BC$ .
The line through $D = ( - 2,6)$ with this normal is $3(x + 2) - 4(y - 6) = 0$ , or $3x - 4y + 30 = 0$ .