From any point on the conic `(x^(2))/(a^(2))+(y^(2))/(b^(2))=4`. Tangents are drawn to the conic `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`. Prove that the normals at the points of contact meet on the conic
`a^(2)x^(2)+b^(2)y^(2)=(1)/(4)(a^(2)-b^(2))^(2)`.

The eccentricity of the conics - (x^(2))/(a^(2)) +(y^(2))/(b^(2)) = 1 is

Tangents at right angle are drawn to the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 . Show that the focus of the middle points of the chord of contact is the curve (x^(2)/a^(2)+y^(2)/b^(2))^(2)=(x^(2)+y^(2))/(a^(2)+b^(2)) .

From any point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 , tangents are drawn to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=2. The area cut-off by the chord of contact on the asymptotes is equal to a/2 (b) a b (c) 2a b (d) 4a b

Two conics (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and x^(2)=-(a)/(b)y intersect, if

Statement 1 : If from any point P(x_1, y_1) on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 , tangents are drawn to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1, then the corresponding chord of contact lies on an other branch of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 Statement 2 : From any point outside the hyperbola, two tangents can be drawn to the hyperbola.

Tangents are drawn from any point on the conic x^2/a^2 + y^2/b^2 = 4 to the conic x^2/a^2 + y^2/b^2 = 1 . Find the locus of the point of intersection of the normals at the point of contact of the two tangents.

Tangents are drawn to hyperbola (x^2)/(16)-(y^2)/(b^2)=1 . (b being parameter) from A(0, 4). The locus of the point of contact of these tangent is a conic C, then:

A normal to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 meets the axes in M and N and lines MP and NP are drawn perpendicular to the axes meeting at P. Prove that the locus of P is the hyperbola a^(2)x^(2)-b^(2)y^(2)=(a^(2)+b^(2))^(2)

the eccentricity to the conic 4x^(2) +16y^(2)-24x-32y=1 is

The line x = at^(2) meets the ellipse x^(2)/a^(2) + y^(2)/b^(2) = 1 in the real points iff