Question

Let BC be affixed line segment in the plane. The locus of a point AA such that the triangle ABC is isosceles, is(with finitely many possible exceptional points).

• A. A line
• B. A circle
• C. The union of a circle and a line
• D. The union of two circles and a line

Answer 1

Answer: (D)

The union of two circles and a line

$3$ cases are possible if $△ABC$ is an isosceles triangle.

Case $1$: If $AB=AC$, then locus of $A$ is a line which is perpendicular to line$BC$ and passing through the midpoint of $BC$

Case$2$: If $AB=BC$, then the locus of $A$ is a circle of radius $AB$ and $B$ as centre of the circle.

Case$2$: If $AC=BC$, then the locus of $A$ is a circle of radius $AC$ and $C$ as centre of the circle.