Question

Let A be a variable point on the line y$=4$. If B and C are variable points on the line $y=-4$ such that triangle ABC is equilateral, then

• A. Perimeter of triangle ABC is $15\sqrt{3}$
• B. Radius of the circle inscribed in triangle ABC is $\frac{4}{3}$
• C. Locus of orthocentre of triangle ABC is $3y+4=0$
• D. $r_1:r_2:r_3=1:1:1$, where $r_1.r_2.r_3$ are ex-radii of triangle ABC.

Answer 1

The correct option is C Locus of orthocentre of triangle ABC is $3y+4=0$

Vertex $A$ of variable equilateral triangle $ABC$ lies on line $y=4$and it's base $BC$ lies on the line $y=-4$.

So ordinate of $A$ is $4$ and ordinates of $B$ and $C$ are $-4$.

Since triangle $ABC$ is equilateral its orthocenter $O$ will coincide with centroid and the ordinate of orthocenter will always be $\frac{1}{3}(4-4-4)=-\frac{4}{3}$

Hence locus of orthocentre is $y=-\frac{4}{3}⟹3y+4=0$