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)`

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)

If the lines joining the origin and the point of intersection of curves ax^(2)+2hxy+by^(2)+2gx+0 and a_(1)x^(2)+2h_(1)xy+b_(1)y^(2)+2g_(1)x=0 are mutually perpendicular,then prove that g(a_(1)+b_(1))=g_(1)(a+b)

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

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

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))

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.