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  • 5. Find the points on the line x + y = 4 which lie at a unit distance from the line 4x + 3y = 10.

Find the points on the line x + y = 4 which lie at a unit distance from the line 4x + 3y = 10.

Answers (1)

Let x1, y1 be any point lying inthe equation x+y=4 

  x1+y1=4…......(1) 

Distance of point x1,y1from the equation 4x+3y=10 

d=\frac{\left | Ax+By+C \right |}{\sqrt{A^{2}+B^{2}}}

1=\frac{4x_{1}+3y_{1}-10}{\sqrt{\left ( 4 \right )^{2}+\left ( 3 \right )^{2}}}=\left | \frac{4x_{1}+3y_{1}-10}{5} \right |

4x1+3y1-10=±5  

 4x1+3y1-10=5 or 4x1+3y1-10=-5

    4x1+3y1=(15)...….(2) or 4x1+3y1=5….....(3)

  From equation 1 we have y1=4-x1….. (4)

 Putting the values of y1in equation 2 we get 4x1+3(4-x1)=15  

  4x1+12-3x1=15 

  x1=15-12 =3

 Putting the value of x1in equation 4  we get y1=1

Now, 4x1+34-x1=5 

   4x1+12-3x1=5

 x1=5-12  x1=-7  

Putting the value in equation 4 we get y1=4--7=4+7=11 

Hence, the required points on the given line are (3,1) and (-7,11) 

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