Question

Let $A$ and $B$ be the fixed points. If the vertex $C$ of triangle $ABC$ moves such that $\cot A+\cot B=$ constant, then the locus of $C$ is

• A. a straight line perpendicular to $AB$
• B. two straight lines parallel to $AB$
• C. a circle
• D. an ellipse

Answer 1

The correct option is B two straight lines parallel to $AB$

Let $AB$ be the base of the triangle.

Let $CD$ be the altitude of the triangle $ABC$.

We have,

$\cot A=\frac{AD}{CD}$ and $\cot B=\frac{BD}{CD}$

$\Rightarrow \cot A+\cot B=\frac{AB}{CD}$

Thus $A$ and $B$ are fixed points.

$\Rightarrow CD=$ constant.

This implies that point $C$ is at a fixed distance from $AB$.

Hence, the locus of $C$ are two parallel lines at a constant distance from $AB$.

Here, we have two parallel lines as $C$ can lie on any side of $AB$.

Hence, option B is correct.