point P(5,2) is equidistant from the point (b,10) and (0,b) .Find b​

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point P(5,2) is equidistant from the point (b,10) and (0,b) .Find b​

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4 weeks 2021-08-21T13:27:06+00:00 1 Answer 1 views 0

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    2021-08-21T13:28:16+00:00

    Given : –

    Point P(5,2) is equidistant from the point (b,10) and (0,b) .

    Required to find : –

    • Value of b ?

    Formula used : –

    The formula to find the distance between any two points is ;

    Distance = √( x₂ – x₁ )² + ( y₂ – y₁ )²

    Solution : –

    Point P(5,2) is equidistant from the point (b,10) and (0,b) .

    We need to find the value of b ?

    So,

    Let the two points which are equidistant from the 2 point be : A & B

    That is ;

    A(b,10) & B(0,b)

    Now,

    From the word equidistant we can conclude that ;

    The distance between these points is equal .

    i.e.

    AP = BP

    So,

    This implies ;

    Let’s try find the distance between AP .

    So,

    The co – ordinates are :

    P(5,2) & A(b,10)

    Here,

    5 = x₁ , 2 = y₁

    b = x₂ , 10 = y₂

    Using the formula ;

    Distance = √( x₂ – x₁ )² + ( y₂ – y₁ )²

    Substitute the values we get ;

    AP = √( b – 5)² + ( 10 – 2 )²

    AP = √( b² + 25 – 10b ) + ( 8 )²

    AP = √b² + 25 – 10b + 64

    AP = √b² – 10b + 89

    Hence,

    The distance between the point A , point P is

    √b² – 10b + 89

    Similarly,

    Now,

    Let’s find the distance between point B & point P

    Here,

    Point B(0,b)

    Point P(5,2)

    x₁ = 5 , y₁ = 2

    x₃ = 0 , y₃ = b

    This implies ;

    The formula becomes as ;

    Distance = √( x₃ – x₁ )² + ( y₃ – y₁ )²

    Substituting the values we get :

    PB = √( 0 – 5 )² + ( b – 2 )²

    PB = √( – 5 )² + ( b² + 4 – 4b )

    PB = √25 + b² + 4 – 4b

    PB = √b² – 4b + 29

    Hence,

    Distance between the point P and point B is

    √b² – 4b + 29 .

    Since,

    AP = BP

    √b² – 10b + 89 = √b² – 4b + 29

    square root get’s cancelled on both sides

    b² – 10b + 89 = b² – 4b + 29

    b² , b² get’s cancelled on both sides

    10b + 89 = 4b + 29

    10b + 4b = 29 89

    6b = 60

    Taking – ( minus ) common on both sides

    ( 6b ) = ( 60 )

    Minus ( – ) get’s cancelled on both sides

    6b = 60

    b = 60/6

    b = 10

    Therefore,

    Value of b = 10 units

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