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Straight Lines and Pair of Straight Lines question

2016 · 10 Apr · Shift 1 · Q36
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Straight Lines and Pair of Straight Lines question

2016 · 10 Apr · Shift 1 · Q36

JEE MainMathematicsStraight Lines and Pair of Straight LinesMCQ+4 / −1
A straight line through origin O meets the lines 3y = 10 − 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio :
  1. A
    2 : 3
  2. B
    1 : 2
  3. C
    4 : 1
  4. D
    3 : 4
View written solutionFree

Correct answer: C

  1. Let the required line through the origin be

y=mxy=mxy=mx

where mmm is its slope.

  1. It meets the line

3y=10−4x3y=10-4x3y=10−4x

at point AAA. Substitute y=mxy=mxy=mx:

3(mx)=10−4x3(mx)=10-4x3(mx)=10−4x 3mx+4x=103mx+4x=103mx+4x=10 x(3m+4)=10x(3m+4)=10x(3m+4)=10

So,

xA=103m+4,yA=m⋅103m+4=10m3m+4x_A=\frac{10}{3m+4}, \qquad y_A=m\cdot \frac{10}{3m+4}=\frac{10m}{3m+4}xA​=3m+410​,yA​=m⋅3m+410​=3m+410m​

Hence,

A(103m+4,10m3m+4)A\left(\frac{10}{3m+4},\frac{10m}{3m+4}\right)A(3m+410​,3m+410m​)

  1. It meets the line

8x+6y+5=08x+6y+5=08x+6y+5=0

at point BBB. Again substitute y=mxy=mxy=mx:

8x+6(mx)+5=08x+6(mx)+5=08x+6(mx)+5=0 x(8+6m)=−5x(8+6m)=-5x(8+6m)=−5

So,

xB=−58+6m,yB=m(−58+6m)=−5m8+6mx_B=-\frac{5}{8+6m}, \qquad y_B=m\left(-\frac{5}{8+6m}\right)=-\frac{5m}{8+6m}xB​=−8+6m5​,yB​=m(−8+6m5​)=−8+6m5m​

Hence,

B(−58+6m,−5m8+6m)B\left(-\frac{5}{8+6m},-\frac{5m}{8+6m}\right)B(−8+6m5​,−8+6m5m​)

  1. Since O=(0,0)O=(0,0)O=(0,0) lies on the same line as AAA and BBB, and divides segment ABABAB, the distances from OOO to AAA and BBB are proportional to the absolute values of their position parameters on the line y=mxy=mxy=mx.

Write points on the line as

(x,y)=t(1,m)(x,y)=t(1,m)(x,y)=t(1,m)

Then for AAA,

tA=103m+4t_A=\frac{10}{3m+4}tA​=3m+410​

and for BBB,

tB=−58+6mt_B=-\frac{5}{8+6m}tB​=−8+6m5​

Therefore,

OA:OB=∣tA∣:∣tB∣=103m+4:58+6mOA:OB=|t_A|:|t_B|=\frac{10}{3m+4}:\frac{5}{8+6m}OA:OB=∣tA​∣:∣tB​∣=3m+410​:8+6m5​

Now simplify:

=10(8+6m):5(3m+4)$$ $$=2(8+6m):(3m+4)$$ Since $8+6m=2(4+3m)=2(3m+4)$, $$2(8+6m):(3m+4)=2\cdot 2(3m+4):(3m+4)=4:1$$ Thus, $$OA:OB=4:1$$ So the origin divides $AB$ in the ratio $$\boxed{4:1}$$ 5. Comparing with options, this is option **C**.
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