Find the condition that the expressions `a x^2-b x y+c y^2a n da_1x^2+b_1x y+c_1y^2` may have factors `y-m xa n dm y-x ,` respectively.

Find the condition that the line A x+B y=1 may be normal to the curve a^(n-1)y=x^ndot

Find the condition if the circle whose equations are x^2+y^2+c^2=2a x and x^2+y^2+c^2-2b y=0 touch one another externally.

Find the equation of the tangent to the ellipse x^2/a^2+y^2/b^2=1 at (x= 1,y= 1) .

Find the linear factors of 2x^2-y^2-x+x y+2y-1.

For what values of KinR the expression 2x^2+Kxy+3y^2-5y-2 can be expressed as (a_1x+b_1y+c_1).(a_2x+b_2y+c_2)

Find the condition if lines x=a y+b ,z=c y+da n dx=a^(prime)y+b^(prime), z=c^(prime)y+d ' are perpendicular.

Factorize: a x+b x+a y+b y +a x^2+b y^2+b x^2+a y^2 a^2+b c+a b+a c + a x-a y+b x-b y

The equations of the perpendicular bisectors of the sides A Ba n dA C of triangle A B C are x-y+5=0 and x+2y=0 , respectively. If the point A is (1,-2) , then find the equation of the line B Cdot

Straight lines y=m x+c_1 and y=m x+c_2 where m in R^+, meet the x-axis at A_1a n dA_2, respectively, and the y-axis at B_1a n dB_2, respectively. It is given that points A_1,A_2,B_1, and B_2 are concylic. Find the locus of the intersection of lines A_1B_2 and A_2B_1 .

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x_1,y_1) and (x_2,y_2), respectively, then (a) x_1=y_2 (b) x_1=y_1 (c) y_1=y_2 (d) x_2=y_1