If the abscissae and the ordinates of two point `Aa n dB` be the roots of `a x^2+b x+c=0a n da^(prime)y^2+b^(prime)y=c^(prime)=0` respectively, show that the equation of the circle described on `A B` as diameter is `a a^(prime)(x^2+y^2)+a^(prime)b x+a b^(prime)y+(c a^(prime)+ca)=0.`

The abscissa and ordinates of the endpoints Aa n dB of a focal chord of the parabola y^2=4x are, respectively, the roots of equations x^2-3x+a=0 and y^2+6y+b=0 . Then find the equation of the circle with A B as diameter.

If (a x^2+b x+c)y+(a^(prime)x^2+b^(prime)x^2+c^(prime))=0 and x is a rational function of y , then prove that (a c^(prime)-a^(prime)c)^2=(a b^(prime)-a^(prime)b)xx(b c^(prime)-b^(prime)c)dot

Fid 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.

Fid 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.

Fid 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.

Show the condition that the curves a x^2+b y^2=1 and a^(prime)\ x^2+b^(prime)\ y^2=1 Should intersect orthogonally

Find the equations to the diagonals of the rectangle the equations of whose sides are x=a ,x=a^(prime),y=b a n d y=b^(prime)

Prove that the line x=a y+b ,\ z=c y+d\ a n d\ x=a^(prime)y+b^(prime),\ z=c^(prime)y+d^(prime) are perpendicular if aa^(prime) + c c^(prime) + 1=0

Prove that the lengths of the perpendiculars from the points (m^2,2m),(m m^(prime),m+m^(prime)), and (m^(prime2),2m^(prime)) to the line x+y+1=0 are in GP.

The condition that one of the straight lines given by the equation a x^2+2h x y+b y^2=0 may coincide with one of those given by the equation a^(prime)x^2+2h^(prime)x y+b^(prime)y^2=0 is (a b^(prime)-a^(prime)b)^2=4(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (a b^(prime)-a^(prime)b)^2=(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (h a^(prime)-h^(prime)a)^2=4(a b^(prime)-a^(prime)b)(b h^(prime)-b^(prime)h) (b h^(prime)-b^(prime)h)^2=4(a b^(prime)-a^(prime)b)(h a^(prime)-h^(prime)a)