If the normals at the four points `(x_1,y_1),(x_2,y_2),(x_3,y_3) and (x_4,y_4)` on the ellipse`x^(2)/a^(2)+y^(2)/b^(2)=1` are concurrent, then the value of `(sum_(i=1)^4x_i)(sum_(i=1)^(4)1/x_i)`

If four points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) and (x_(4),y_(4)) taken in order in a parallelogram, then:

Let (x_(1), y_(1)), (x_(2),y_(2)), (x_(3),y_(3)) and (x_(4), y_(4)) are four points which are at unit distance from the lines 3x-4y+1=0 and 8x+6y+1=0, then the value of (Sigma_(i=1)^(4)x_(i))/(Sigma_(i=1)^(4)y_(i)) is equal to

If the points (x_1, y_1),(x_2,y_2), and (x_3, y_3) are collinear show that (y_2-y_3)/(x_2x_3)+(y_3-y_1)/(x_3x_1)+(y_1-y_2)/(x_1x_2)=0

If P(x_1,y_1),Q(x_2,y_2) and R(x_3, y_3) are three points on y^2 =4ax and the normal at PQ and R meet at a point, then the value of (x_1-x_2)/(y_3)+(x_2-x_3)/(y_1)+(x_3-x_1)/(y_2)=

If the normals at (x_(1),y_(1)) and (x_(2) ,y_(2)) on y^(2) = 4ax meet again on parabola then x_(1)x_(2)+y_(1)y_(2)=

If the normal at the point P(x_1y_1),i=1.2,3,4 on the hyperbola xy=c^2 are concurrent at the point Q(h, k), then ((x_1+x_2+x_3+x_4)(y_1+y_2+y_3+y_4))/(x_1x_2x_3x_4) is:

If the normals to the ellipse x^2/a^2+y^2/b^2= 1 at the points (x_1, y_1), (x_2, y_2) and (x_3, y_3) are concurrent, prove that |(x_1,y_1,x_1y_1),(x_2,y_2,x_2y_2),(x_3,y_3,x_3y_3)|=0 .

If the line y=x+sqrt(3) touches the ellipse (x^(2))/(4)+(y^(2))/(1)=1 then the point of contact is

If the hyperbola xy=c^(2) intersects the circle x^(2)+y^(2)=a^(2)" is four points "P(x_(1),y_(1)), Q(x_(2),y_(2)), R(x_(3),y_(3)) and S(x_(4),y_(4)) then show that (i) x_(1)+x_(2)+x_(3)+x_(4)=0 (ii) y_(1)+y_(2)+y_(3)+y_(4)=0 (iii) x_(1)x_(2)x_(3)x_(4)=c^(4) (iv) y_(1)y_(2)y_(3)y_(4)=c^(4)

If the normals at four points P (x_i y_i), i = 1, 2, 3, 4 on the rectangular hyperbola xy = c^2 , meet at the point Q(h, k), then prove that x_1 + x_2 + x_3 + x_4 =h