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If by change of axes without change of o...

If by change of axes without change of origin, the expression `ax^(2)+2hxy+by^(2)` becomes `a_(1)x_(1)^(2)+2h_(1)x_(1)y_(1)+b_(1)y_(1)^(2)`, prove that
`(a-b)^(2)+4h^(2)=(a_(1)-b_(1))^(2)+4h_(1)^(2)`

Answer

Step by step text solution for If by change of axes without change of origin, the expression ax^(2)+2hxy+by^(2) becomes a_(1)x_(1)^(2)+2h_(1)x_(1)y_(1)+b_(1)y_(1)^(2), prove that (a-b)^(2)+4h^(2)=(a_(1)-b_(1))^(2)+4h_(1)^(2) by MATHS experts to help you in doubts & scoring excellent marks in Class 12 exams.

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Find the condition that the expressions ax^(2)-bxy+cy^(2) and a_(1)x^(2)+b_(1)xy+c_(1)y^(2) may have factors y-mx and my-x respectively.

If one of the roots of the equation ax^(2)+bx+c=0 be reciprocal of one of the a_(1)x^(2)+b_(1)x+c_(1)=0, then prove that (aa_(1)-cc_(1))^(2)=(bc_(1)-ab_(1))(b_(1)c-a_(1)b)

Knowledge Check

  • The lines joining the origin to the points of intersection of the curve ax^(2)+2hxy+by^(2)+2gx=0 and a_(1)x^(2)+2b_(1)xy+b_(1)y^(2)+2g_(1)x=0 are _|_ then

    A
    `(a+b)/(g_(1))=(a_(1)+b_(1))/g`
    B
    `(a+b)g_(1)=(a_(1)+b_(1))g`
    C
    `(a-b)g=(a_(1)-b_(1))g_(1)`
    D
    none of these

  • the value of the determinant |{:((a_(1)-b_(1))^(2),,(a_(1)-b_(2))^(2),,(a_(1)-b_(3))^(2),,(a_(1)-b_(4))^(2)),((a_(2)-b_(1))^(2),,(a_(2)-b_(2))^(2) ,,(a_(2)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(3)-b_(1))^(2),,(a_(3)-b_(2))^(2),,(a_(3)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(4)-b_(1))^(2),,(a_(4)-b_(2))^(2),,(a_(4)-b_(3))^(2),,(a_(4)-b_(4))^(2)):}| is

    A
    dependant on `a_(i),i=1,2,3,4`
    B
    dependant on `b_(i),i=1,2,3,4`
    C
    dependant on `a_(ij), b_(i) i= 1,2,3,4`
    D
    0

  • Statement 1 : If two conics a_(1)x^(2)+ 2h_(1)xy+b_(1)^(2)=c_(1) , a_(2)x^(2) +2h_(2) xy +b_(2)y^(2) =c_(2) intersect in 4 concyclic points, then ( a_(1) -b_(1)) h_(2)=(a_(2)-b_(2))h_(1) . Statement 2 : For a conic to be a circle, coefficient of x^(2) = coefficient of y^(2) and coefficient of xy =0.

    A
    Statement-1 `:` is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
    B
    Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement -1
    C
    Statement-1 is True, Statement -2 is False
    D
    Statement-1 is False, Statement-2 is True

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    If the points (a_(1),b_(1))m(a_(2),b_(2))" and " (a_(1)-a_(2),b_(2)-b_(2)) are collinear, then prove that a_(1)/a_(2)=b_(1)/b_(2)

    Show that the circle passing through the origin and cutting the circles x^(2)+y^(2)-2a_(1)x-2b_(1)y+c_(1)=0 and x^(2)+y^(2)-2a_(2)x-2b_(2)y+c_(2)=0 orthogonally is det[[c_(1),a_(1),b_(2)c_(1),a_(1),b_(2)c_(2),a_(2),b_(2)]]=0

    If the system of equations a_(1)x+b_(1)y+c_(1),a_(2)x+b_(2)y+c_(2)=0 is inconsistent,(a_(1))/(a_(2))=(b_(1))/(b_(2))!=(c_(1))/(c_(2))

    Area of the parallelogram formed by the lines a_(1)x+b_(1)y+c_(1)=0,a_(1)x+b_(1)y+d_(1)=0 and a_(2)x+b_(2)y+c_(2)=0,a_(2)x+b_(2)y+d_(2)=0 is ([d_(1)-c_(1))(d_(2)-c_(2)))/([(a_(1)^(2)+b_(1)^(2))(a_(2)^(2)+b_(2)^(2))]^(1//2))

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