FIND THE RATIO THAT THE POINT DIVIDES THE LINE SEGMENT

Problem 1 :

Find the ratio in which the point (11, 15) divides the line segment joining the point (15, 5) and (9, 20).

Solution:

Given points are (15, 5) and (9, 20).

Let P(11, 15) divide AB internally in the ratio l : m.

By the section formula,

P(x,y)=lx2+mx1l+m,ly2+my1l+mSubstitute (x1, y1)=(15,5) and (x2, y2)=(9, 20)(11,15)=l(9)+m(15)l+m,l(20)+m(5)l+m(11,15)=9l+15ml+m,20l+5ml+m

Equating the coefficients of x, we get

9l+15ml+m=119l+15m=11(l+m)9l+15m=11l+11m2l=4mlm=42l:m=4:2

Hence, the point divides the line segment joining the points in the ratio 4 : 2.

Problem 2 :

Find the ratio in which the point P(m, 6) divides the line segment joining the point A(-4, 3) and B(2, 8). Also find the value of m.

Solution:

Using the section formula, if a point (x, y) divided the line joining the points (x1, y1) and (x2, y2) in the ratio l : m, then

(x,y)=lx2+mx1l+m,ly2+my1l+m

Let P divides the join of A and B in the ratio k : 1, then

Therefore, we have

6=k×8+1×3k+16(k+1)=8k+36k+6=8k+32k=3k=32P divides the join of A and B in the ratio 3 : 2Now, m=2k-4k+1=2×32-432+1=-152m=-25

Problem 3 :

Find the ratio in which the point P(-6, a) divides the join of A(-3, -1) and B(-8, 9). Also find the value of a.

Solution:

Using the section formula, if a point (x, y) divided the line joining the points (x1, y1) and (x2, y2) in the ratio l : m, then

P(x,y)=lx2+mx1l+m,ly2+my1l+mSubstitute (x1, y1)=(-3,-1) and (x2, y2)=(-8,9)P(-6,a)=l(-8)+m(-3)l+m,l(9)+m(-1)l+mP(-6,a)=-8l-3ml+m,9l-ml+mEquating the x component-6=-8l-3ml+m-6(l+m)=-8l-3m-6l-6m=-8l-3m2l=3mlm=32

l : m = 3 : 2

To find a, equating the coefficient of y

a=9l-ml+ma=9(3)-23+2a=255a=5

Problem 4 :

Find the ratio in which the point P(-3, a) divides the join of A(-5, -4) and B(-2, 3). Also find the value of a.

Solution:

Using the section formula, if a point (x, y) divided the line joining the points (x1, y1) and (x2, y2) in the ratio l : m, then

P(x,y)=lx2+mx1l+m,ly2+my1l+mSubstitute (x1, y1)=(-5,-4) and (x2, y2)=(-2,3)P(-3,a)=l(-2)+m(-5)l+m,l(3)+m(-4)l+mP(-3,a)=-2l-5ml+m,3l-4ml+mEquating the x component-3=-2l-5ml+m-3(l+m)=-2l-5m-3l-3m=-2l-5m-l=2mlm=21l:m=2:1

To find a, equating the coefficient of y

a=3l-4ml+ma=3(2)-4(1)2+1a=6-43a=23

Problem 5 :

Find the ratio in which the point P(a, 1) divides the join of A(-4, 4) and B(6, -1). Also find the value of a.

Solution:

Using the section formula, if a point (x, y) divided the line joining the points (x1, y1) and (x2, y2) in the ratio l : m, then

P(x,y)=lx2+mx1l+m,ly2+my1l+mSubstitute (x1, y1)=(-4,4) and (x2, y2)=(6,-1)P(a,1)=l(6)+m(-4)l+m,l(-1)+m(4)l+mP(a,1)=6l-4ml+m,-l+4ml+mEquating the y component1=-l+4ml+m1(l+m)=-l+4ml+l=4m-m2l=3mlm=32l:m=3:2

To find a, equating the coefficient of x

a=6l-4ml+ma=6(3)-4(2)3+2a=18-85a=2

Problem 6 :

Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by (-1, 6).

Solution:

P(x,y)=lx2+mx1l+m,ly2+my1l+mSubstitute (x1, y1)=(-3,10) and (x2, y2)=(6,-8)P(-1,6)=l(6)+m(-3)l+m,l(-8)+m(10)l+mP(-1,6)=6l-3ml+m,-8l+10ml+mEquating the coefficient of x, we get-1=6l-3ml+m-1(l+m)=6l-3m-l-m=6l-3m-7l=-2mlm=27l:m=2:7

Hence, the point divides the line segment joining the points in the ratio 2 : 7.

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