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=a_(1)+b_(1)`

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)

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)

If two circles x^(2)+y^(2)+2a_(1)x+2b_(1)y=0 and x^(2)+y^(2)+2a_(2)x+2b_(2)y=0 touches then show that a_(1)b_(2)=a_(2)b_(1)

Prove that the equation of tangent of the ellipse x^(2)/a^(2)+y^(2)/b^(2) = 1" at point "(x_(1),y_(1))" is (xx_(1))/a^(2) + (yy_(1))/b^(2)=1 .

If the lines x=a_(1)y + b_(1), z=c_(1)y +d_(1) and x=a_(2)y +b_(2), z=c_(2)y + d_(2) are perpendicular, then

If (x - a) be a factor common to a_(1)x^(2) + b_(1)x +c and a_2x^2 + b_(2)x + c , then prove that: alpha(a_(1)-a_(2))=b_(2)-b_(1)

Statement - 1 : For the straight lines 3x - 4y + 5 = 0 and 5x + 12 y - 1 = 0 , the equation of the bisector of the angle which contains the origin is 16 x + 2 y + 15 = 0 and it bisects the acute angle between the given lines . statement - 2 : Let the equations of two lines be a_(1) x + b_(1) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 where c_(1) and c_(2) are positive . Then , the bisector of the angle containing the origin is given by (a_(1) x + b_(1) y + c_(1))/(sqrt(a_(2)^(2) + b_(1)^(2))) = (a_(2) x + b_(2)y + c_(2))/(sqrt(a_(2)^(2) + b_(2)^(2))) If a_(1) a_(2) + b_(1) b_(2) gt 0 , then the above bisector bisects the obtuse angle between given lines .

If omega is a non-real cube root of unity, then Delta = |(a_(1) + b_(1) omega,a_(1) omega^(2) + b_(1),a_(1) + b_(1) + c_(1) omega^(2)),(a_(2) + b_(2) omega,a_(2) omega^(2) + b_(2),a_(2) + b_(2) + c_(2) omega^(2)),(a_(3) + b_(3) omega,a_(3) omega^(2) + b_(3),a_(3) + b_(3) + c_(3) omega^(2))| is equal to

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.

If a_(1) x^(3) + b_(1)x^(2) + c_(1)x + d_(1) = 0 and a_(2)x^(3) + b_(2)x^(2) + c_(2)x + d_(2) = 0 a pair of repeated roots common, then prove that |{:(3a_(1)", "2b_(1) ", "c_(1)),(3a_(2)", " 2b_(2)", "c_(2)),(a_(2)""b_(1)- a_(1)b_(2)", "c_(2)a_(1)-c_(2)a_(1)", "d_(1)a_(2)-d_(2)a_(1)):}|=0