Home
Class 12
MATHS
A(x1,y1), B(x2,y2),C(x3,y3) are three ve...

`A(x_1,y_1), B(x_2,y_2),C(x_3,y_3)` are three vertices of a triangle ABC, `lx+my+n=0` is an equation of line L. If L intersects the sides BC,CA and AB of a triangle ABC at P,Q,R respectively, then `(BP)/(PC)xx(CQ)/(QA)xx(AR)/(RB)` is equal to

A

`-1`

B

`-(1)/(2)`

C

`(1)/(2)`

D

1

Text Solution

Verified by Experts

The correct Answer is:
A
Let `BP: PC = alpha: 1`
`:. P ((alpha x_(3)+x_(2))/(alpha+1),(alphay_(3)+y_(2))/(alpha+1))`
P lies on `lx +my +n = 0`
`:. L ((alpha x_(3)+x_(2))/(alpha+1)) +m ((alphay_(3)+y_(2))/(alpha+1)) +n = 0`
`:. alpha =(-(lx_(2)+my_(2)+n))/((lx_(3)+my_(3)+n))`
Similarly, if `CQ: QA = beta:1`
`:. beta = (-(lx_(3)+my_(3)+n))/((lx_(1)+my_(1)+n))`
and if `AR: RB = gamma:1`
`:. gamma =(-(lx_(1)+my_(1)+n))/((lx_(2)+my_(2)+n))`
`:. (BP)/(PC).(CQ)/(QA).(AR)/(RB) =-1`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • COORDINATE SYSTEM

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|2 Videos

  • COORDINATE SYSTEM

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|2 Videos

  • CONTINUITY AND DIFFERENTIABILITY

    CENGAGE PUBLICATION|Exercise All Questions|238 Videos

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise JEE Main|6 Videos

Similar Questions

Explore conceptually related problems

A straight line L interesect the sides BC, CA and BA of triangleABC at P,Q and R resp. Prove that (PR)/(PQ).(CQ)/(AC).(AB)/(RB) = 1 .

A(x_(1),y_(1)), B(x_(2),y_(2)), C(x_(3),y_(3)) are three vertices of a triangle ABC. lx +my +n = 0 is an equation of the line L. If P divides BC in the ratio 2:1 and Q divides CA in the ratio 1:3 then R divides AB in the ratio (P,Q,R are the points as in problem 1)

Knowledge Check

  • A(x_1,y_1),B(x_2,y_2),C(x_3,y_3) are the vertices of a triangle ABC. lx+my+n=0 is an equation of the line L. Answer the following questions based on above passage : If L intersects the sides BC, CA and AB of the triangle ABC at P, Q, R respectively then (BP)/(PC)xx(CQ)/(QA)xx(AR)/(RB) is equal to

    A
    -1
    B
    `-1//2`
    C
    `1//2
    D
    1

  • A(x_1,y_1),B(x_2,y_2),C(x_3,y_3) are the vertices of a triangle ABC. lx+my+n=0 is an equation of the line L. Answer the following questions based on above passage : If the centroid of the triangle ABC is at the origin and algebraic sum of the length of the perpendiculars from the vertices of the triangle ABC on the line L is equal to 1 then sum of the squares of the intercepts made by L on the coordinate axes is equal to

    A
    0
    B
    4
    C
    9
    D
    16

  • A(a, b), B(x_1, y_1) and C(x_2, y_2) are the vertices of a triangle. If a, x_1, x_2 are in GP with common ratio r and b, y_1,y_2 are in GP with common ratio s, then area of triangle ABC is :

    A
    ab(r-1)(s-1)(s-r)
    B
    1/2 ab(r+1)(s+1)(s-r)
    C
    1/2 ab(r-1)(s-1)(s-r)
    D
    ab(r+1)(s+1)(r-s)

    • Similar Questions

    Explore conceptually related problems

    A(0,sqrt(2))andB(2sqrt(2),0) are two vertices of a triangle ABC and AB=BC . If the equation of the side BC be x=2sqrt(2) , then the area of triangle ABC is (in sq . Units )-

    A(2, 3, 1), B(-2, 2, 0) and C(0, 1, -1) are the vertices of the triangle ABC. Show that the triangle ABC is right -angled.

    (4, -3), (-5, 2) and (x, y) are the three vertices of a triangle. If the centroid of the triangle be the origen, find the values of x and y.

    The equations of the perpendicular bisectors of the sides AB and AC of a triangle ABC are x -y+5=0 and x+2y=0 respectively . If the point A is (1,-2) , find the equation of the line BC.

    The midpoints of AB and AC of a triangle ABC are respectively X and Y . If BC+XY= 12 units then BC-XY=

    Home
    Profile