In a ΔABC , A = (1,2) ; B =(5,5) In angleACB = 90° If area of ΔABC is to be 6.5 squnits the possible number of points for C are <

In a ΔABC , A = (1,2) ; B =(5,5) In angleACB = 90° If area of ΔABC is to be 6.5 squnits the possible number of points for C are

Options

A ) 1
B) 2
C) 0
D) 4​

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  1. Topic :-

    Coordinate Geometry

    Given :-

    In a ΔABC, A ≡ (1, 2); B ≡ (5, 5) and ∠ACB = 90°.

    Area of ΔABC is to be 6.50 sq. units.

    To Find :-

    Possible number of points for C.

    Solution :-

    ΔABC is a right angle triangle as ∠ACB = 90°.

    Side AB opposite to ∠ACB will act as hypotenuse for the triangle.

    Calculating length AB from Distance Formula,

    [tex]AB=\sqrt{(5-1)^2+(5-2)^2}\;units[/tex]

    [tex]AB=\sqrt{4^2+3^2}\;units[/tex]

    [tex]AB=\sqrt{16+9}\;units[/tex]

    [tex]AB=\sqrt{25}\;units[/tex]

    [tex]AB=5\;units[/tex]

    Assuming length of arms of triangle,

    Let length of arms of given triangle be x and y.

    Applying Pythagoras Theorem,

    x² + y² = 5² . . . . equation (1)

    Area of Right Angle Triangle,

    [tex]Area=\dfrac{1}{2} \times (Product\:of\:length\:of\:arms)[/tex]

    [tex]6.5=\dfrac{1}{2} \times x \times y[/tex]

    [tex]13=xy[/tex]

    [tex]x=\dfrac{13}{y}[/tex]

    Substituting value of ‘x’ in equation (1),

    [tex]x^2+y^2=5^2[/tex]

    [tex]\left( \dfrac{13}{y} \right)^2+y^2=5^2[/tex]

    [tex]\dfrac{169}{y^2} +y^2=25[/tex]

    [tex]\dfrac{169+y^4}{y^2}=25[/tex]

    Cross Multiply,

    [tex]169+y^4=25y^2[/tex]

    Rearranging it,

    [tex]y^4-25y^2+169=0[/tex]

    Substitite y² = t,

    [tex](y^2)^2-25y^2+169=0[/tex]

    [tex]t^2-25t+169=0[/tex]

    Calculating value of Discriminant,

    [tex]D=b^2-4ac[/tex]

    Here,

    a = 1

    b = -25

    c = 169

    [tex]D=(-25)^2-4(1)(169)[/tex]

    [tex]D=625-676[/tex]

    [tex]D=-51[/tex]

    [tex]D<0[/tex]

    which means

    Real ‘t’ doesn’t exist which means Real ‘y’ doesn’t exist.

    Thus, there are no possible point for point C as y doesn’t exist.

    Answer :-

    So, there are Zero (0) possible points for point C.

    Hence, option C is correct.

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