Question

A and b are two fixed points . The vertex c of a triangle abc moves such that cota + cotb = constant. Locus of c is a straigt line

Answer 1

Locus of c is a straight line Parallel to AB

Step-by-step explanation:

Let say A = (-a , 0)  & B = (a , 0)

AB is  y = 0

C = ( x , y)

Draw CD ⊥ AB  

=> D = ( x , 0)

CD = y

AD = a + x    BD = a - x

CotA = (a + x)/y

CotB = (a - x)/y

CotA + CotB = K  ( constant)

=>  (a + x)/y + (a - x)/y = K

=> 2a = Ky

=> y  = 2a/K

=> This is a straight line Parallel to AB at a distance of 2a/K

Locus of c is a straight line Parallel to AB

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Answer 2

Locus of c is a straight line.

Step-by-step explanation:

Given, ΔABC in which points A and B are fixed.

cotA + cotB = constant

Construction: Draw line AD such that point D lie on BC.

We have,

$cot A=\frac{AD}{CD}$

$cot B =\frac{DB}{CD}$

Now, $cot A+cot B = \frac{AD}{CD}+\frac{DB}{CD}$

                             =$\frac{AD+DB}{CD}$

                             = $\frac{AB}{CD}$

  $constant = \frac{AB}{CD}$

Since,A and B are fixed points .

⇒AB is constant

If AB is constant

⇒CD is also constant.

Hence,locus of c is a straight line which is parallel to AB.