The tangent at P meets the axes in T and t and CY is the perpendicular on it from the centre; prove that (1) `Tt *PY=a^2-b^2 `

Find the points on the ellipse x^(2)+3y^(2)=6 where the tangent are equally inclined to the axes.Prove also that the length of the perpendicular from the centre on either of these tangents is 2 .

Find the coordinates of those points on the ellipse x^2/a^2 + y^2/b^2 = 1 , tangents at which make equal angles with the axes. Also prove that the length of the perpendicular from the centre on either of these is sqrt((1/2)(a^2+b^2)

Consider an ellipse x^2/25+y^2/9=1 with centre c and a point P on it with eccentric angle pi/4. Nomal drawn at P intersects the major and minor axes in A and B respectively. N_1 and N_2 are the feet of the perpendiculars from the foci S_1 and S_2 respectively on the tangent at P and N is the foot of the perpendicular from the centre of the ellipse on the normal at P. Tangent at P intersects the axis of x at T.

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 tangent at any point P of a circle x^(2)+y^(2)=a^(2) meets the tangent at a fixed point A(a,0) in T and T is joined to B ,the other end of the diameter through A ,prove that the locus of the intersection of AP and BT is an ellipse whose eccentricity is (1)/(sqrt(2))

A tangent to 3x^2 + 4y^2=12 is equally inclined with the coordinate axes. Then the perpendicular distance from the centre of the ellipse to this tangent is

From any point P tangents of length t_(1) and t_(2) are drawn to two circles with centre A,B and if PN is the perpendicular from P to the radical axis and t_(1)^(2) - t_(2)^(2) = K.PN*AB then K=

The tangent at P on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 cuts the major axis in T and PN is the perpendicular to the x -axis, C being centre then CN.CT