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The vertices of a triangle are [at(1)t(...

The vertices of a triangle are
`[at_(1)t_(2),a(t_(1)+t_(2))],[at_(2)t_(3),a(t_(2)+t_(3))]`
`[at_(3)t_(1),a(t_(3)+t_(1))]`
Find the coordinates of its orthocentre.

Text Solution

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The correct Answer is:
To find the orthocenter of the triangle with vertices given by the coordinates \((at_1t_2, a(t_1 + t_2))\), \((at_2t_3, a(t_2 + t_3))\), and \((at_3t_1, a(t_3 + t_1))\), we will follow these steps: ### Step 1: Identify the vertices of the triangle Let the vertices of the triangle be: - \( A = (at_1t_2, a(t_1 + t_2)) \) - \( B = (at_2t_3, a(t_2 + t_3)) \) - \( C = (at_3t_1, a(t_3 + t_1)) \)
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Explore conceptually related problems

The vertices of a triangle are [at_(1)t_(2),a(t_(1)+t_(2))],[at_(2),a(t_(2)+t_(3))] and [at_(3)t_(1),a(t_(3)+t_(1))] . The co ordinates of the orthocentre of the triangle are…….

The vertices of a triangle are [at_(1)t_(2),a(t_(1)+t_(2))] , [at_(2)t_(3),a(t_(2)+t_(3))] , [at_(3)t_(1),a(t_(3)+t_(1))] . Find the orthocentre of the triangle.

Knowledge Check

  • If O is the orthocentre of triangle ABC whose vertices are at A(at_(1)^(2),2at_(1), B (at_(2)^(2),2at_(2)) and C (at_(3)^(2), 2at_(3)) then the coordinates of the orthocentreof Delta O'BC are

    A
    `(a(t_(1)^(2)+t_(2)^(2)+t_(3)^(2)),2a(t_(1)+t_(2)+t_(3)))`
    B
    `(-a,0)`
    C
    `(at_(1)^(2),2at_(1))`
    D
    `(0,a)`

  • If t_(1),t_(2),t_(3) are distinct then the points (t_(1),2at_(1)+at_(1)^(3),(t_(2),2at_(2)+at_(2)^(3)) and (t_(3),2at_(3)+at_(3))^(3)) are collinear if

    A
    `t_(1)t_(2)t_(3)=-1`
    B
    `t_(1)+t_(2)+t_(3)=t_(1)t_(2)t_(3)`
    C
    `t_(1)+t_(2)+t_(3)=0`
    D
    `t_(1)+t_(2)+t_(3)=-1`

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    The vertices of a triangle are [a t_1t_2,a(t_1 +t_2)], [a t_2t_3,a(t_2 +t_3)], [a t_3t_1,a(t_3 +t_1)] Then the orthocenter of the triangle is (a) (-a, a(t_1+t_2+t_3)-at_1t_2t_3) (b) (-a, a(t_1+t_2+t_3)+at_1t_2t_3) (c) (a, a(t_1+t_2+t_3)+at_1t_2t_3) (d) (a, a(t_1+t_2+t_3)-at_1t_2t_3)

    Find the area of that triangle whose vertices are (at_(1)^(2),2at_(1)),(at_(2)^(2),2at_(2))and(at_(3)^(2),2at_(3)).

    Find the are of triangle whose vertex are: (at_(1),(a)/(t_(1)))(at_(2),(a)/(t_(2)))(at_(3),(a)/(t_(3)))

    Find the coordinates o the vertices of a triangle,the equations of whose sides are: y(t_(1)+t_(2))=2x+2at_(1)at_(2),y(t_(2)+t_(3))=2x+2at_(2)t_(3) and ,y(t_(3)+t_(1))=2x+2at_(1)t_(3)

    Prove that the area of the triangle whose vertices are : (at_(1)^(2),2at_(1)) , (at_(2)^(2),2at_(2)) , (at_(3)^(2),2at_(3)) is a^(2)(t_(1)-t_(2))(t_(2)-t_(3))(t_(3)-t_(1)) .

    Calculate a, T_(1), T_(2), T_(1)' & T_(2)' .

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