If `A` is the centre of the circle, `x^2 + y^2 +2g_i x + 5 = 0` and `t_i` is the length of the tangent from any point to this circle, `i = 1, 2, 3,` then show that `(g_2 - g_3) t_1^2 +(g_3- g_1)t_2^2 + (g_1 - g_2) t_3^2 = 0`

If the length of the tangent from (f,g) to the circle x^(2)+y^(2)=6 be twice the length of the tangent from the same point to the circle x^(2)+y^(2)+3x+3y=0, then

If the length of the tangent drawn from (f, g) to the circle x^2+y^2= 6 be twice the length of the tangent drawn from the same point to the circle x^2 + y^2 + 3 (x + y) = 0 then show that g^2 +f^2 + 4g + 4f+ 2 = 0 .

If the length of the tangent from a point (f,g) to the circle x^2+y^2=4 be four times the length of the tangent from it to the circle x^(2)+y^(2)=4x , show that 15f^(2)+15g^(2)-64f+4=0

Length of the tangent. Prove that the length t o f the tangent from the point P (x_(1), y(1)) to the circle x^(2) div y^(2) div 2gx div 2fy div c = 0 is given by t=sqrt(x_(1)^(2)+y_(1)^(2)+2gx_(1)+2fy_(1)+c) Hence, find the length of the tangent (i) to the circle x^(2) + y^(2) -2x-3y-1 = 0 from the origin, (2,5) (ii) to the circle x^(2)+y^(2)-6x+18y+4=-0 from the origin (iii) to the circle 3x^(2) + 3y^(2)- 7x - 6y = 12 from the point (6, -7) (iv) to the circle x^(2) + y^(2) - 4 y - 5 = 0 from the point (4, 5).

If g(x) = (x^3 - ax)/(4) , and g(2) = 1/2 , what is the value of g(4) ?

If G_1 is the mean centre of A_1,B_1,C_1 and G_2 that of A_2,B_2,C_2 then show that vec(A_1A_2)+vec(B_1B_2)+vec(C_1C_2)=3vec(G_1G_2)

Prove that the general equation of circles cutting the circles x^2 + y^2 + 2g_r x + 2f_r y+c_r = 0, r= 1, 2 orthogonally is : |(x^2+y^2,-x,-y),(c_1,g_1,f_1),(c_2,g_2,f_2)|+k|(-x, -y, 1), (g_1, f_1, 1), (g_2, f_2, 1)| =0

Tangents and normal from a point (3, 1) to circle C whose equation x^(2) + y^(2) - 2x - 2y + 1 = 0 Let the points of contact of tangents be T_(i)(x_(i),(y_(i)), where I = 1, 2 and feet of normal be N_(1) and N_(2)( N_(1) " is near" P) Tangents ared drawn at N_(1) and N_(2) and normals are drawn at T_(1) and T_(2)

Statement 1 : If a circle S=0 intersects a hyperbola x y=4 at four points, three of them being (2, 2), (4, 1) and (6,2/3), then the coordinates of the fourth point are (1/4,16) . Statement 2 : If a circle S=0 intersects a hyperbola x y=c^2 at t_1,t_2,t_3, and t_3 then t_1-t_2-t_3-t_4=1

Let f (x) = [{:(1-x,,, 0 le x le 1),(0,,, 1 lt x le 2 and g (x) = int_(0)^(x) f (t)dt.),( (2-x)^(2),,, 2 lt x le 3):} Let the tangent to the curve y = g( x) at point P whose abscissa is 5/2 cuts x-axis in point Q. Let the prependicular from point Q on x-axis meets the curve y =g (x) in point R .Find equation of tangent at to y=g(x) at P .Also the value of g (1) =