Show that the length of the least chord of the circle `x^2+y^2+2gx+2fy+c=0` which passes through an internal point `(alpha, beta)` is equal to `2sqrt(-(alpha^2+beta^2+2galpha+2fbeta+c))`.

(a) show that alpha^2 + beta^2 + alpha beta =0

Prove that the locus of the middle-points of the chords of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 which pass through a fixed point (alpha, beta) is a hyperbola whose centre is ((alpha)/(2), (beta)/(2)) .

The line ax+by+by+c=0 is normal to the circle x^(2)+y^(2)+2gx+2fy+d=0, if

I If a point (alpha, beta) lies on the circle x^2 +y^2=1 then the locus of the point (3alpha.+2, beta), is

Find the locus of the mid-point of the chords of the circle x^2 + y^2 + 2gx+2fy+c=0 which subtend an angle of 120^0 at the centre of the circle.

The area of an equilateral triangle inscribed in the circle x^(2)+y^(2)+2gx+2fy+c=0 is

A variable chord of circle x^(2)+y^(2)+2gx+2fy+c=0 passes through the point P(x_(1),y_(1)) . Find the locus of the midpoint of the chord.

A chord through P cut the circle x^(2)+y^(2)+2gx+2fy+c=0 in A and B another chord through P in c and D, then

Show that the circle S-= x^(2) + y^(2) + 2gx + 2fy + c = 0 touches the (i) X- axis if g^(2) = c

The chrods of contact of the pair of tangents to the circle x^(2)+y^(2)=1 dravwm from any point on the line 2x+y=4 paas through the point (alpha,beta) then find alpha and beta .