Question

If abc are unequal and different from 1 such that the points  are collinear, then  

Solution

Correct option is

bc + ca + ab – abc = 3 (a + b + c)

Suppose the given points lie on the line

lx + my + n = 0

then a, b, c are the roots of the equation.  

        

⇒                    a + b + c = –m/l  

                       bc + ca + ab = n/l  

                       abc = (3m + n)/  

Eliminating lmn, we get

              abc = –3(a + b + c) + bc + ca + ab  

⇒          bc + ca + ab – abc = 3(a + b + c)

SIMILAR QUESTIONS

Q1

The number of integral points (integral point means both the coordinates should be integer) that lie exactly in the interior of the triangle with vertices (0, 0), (0, 21), and (21, 0) is

Q2

Let P = (–1, 0), Q = (0, 0) and  be three points. Then the equation of the bisector of the angle PQR is 

Q3

A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. Then the point O divides the segment PQ in the ratio.

Q4

Let A0A1 2 A3 A4 A5 be a regular hexagon described in a circle of unit radius. Then the product of the length of the line segments A A1A0 A2and A0 A4 is 

Q5

 

The diagonals of a parallelogram PQRS are long the lines

x + 3y = 4 and 6x – 2y = 7, then PQRS must be a

Q6

The orthocenter of the triangle formed by the lines xy = 0 and x + y = 1 is 

Q7

The straight lines x + y = 0, 3x + y – 4 = 0, x + 3y – 4 = 0 form a triangle which is

Q8

If sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is

Q9

If the circumcentre of a triangle lies at the origin and centroid is the middle point of the line joining the points (a2 + 1, a2 + 1) and (2a, –2a), then the orthocenter lies on the line.

Q10

If two vertices of a triangle are (–2, 3) and (5, –1), orthocentre lies at the origin and centroid on the line x + y = 7, then the third vertex lies at