A normal to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` meets the axes at `Ma n dN` and lines `M P` and `N P` are drawn perpendicular to the axes meeting at `Pdot` Prove that the locus of `P` is the hyperbola `a^2x^2-b^2y^2=(a^2+b^2)^2 dot`

For the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , let n be the number of points on the plane through which perpendicular tangents are drawn.

Two perpendicular tangents to the circle x^2 + y^2= a^2 meet at P. Then the locus of P has the equation

If the normal at P(theta) on the hyperbola (x^2)/(a^2)-(y^2)/(2a^2)=1 meets the transvers axis at G , then prove that A GdotA^(prime)G=a^2(e^4sec^2theta-1) , where Aa n dA ' are the vertices of the hyperbola.

The tangent at a point P on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 passes through the point (0,-b) and the normal at P passes through the point (2asqrt(2),0) . Then the eccentricity of the hyperbola is

For the hyperbola (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 , the normal at point P meets the transverse axis AA' in G and the connjugate axis BB' in g and CF be perpendicular to the normal from the centre. Q. Locus of middle-point of G and g is a hyperbola of eccentricity

If the normal at any point P on the ellipse x^2/a^2+y^2/b^2=1 meets the axes at G and g respectively, then find the ratio PG:Pg .

Length of common tangents to the hyperbolas x^2/a^2-y^2/b^2=1 and y^2/a^2-x^2/b^2=1 is

For the hyperbola (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 , the normal at point P meets the transverse axis AA' in G and the connjugate axis BB' in g and CF be perpendicular to the normal from the centre. Q. The value PF*Pg is equal to

P is a point on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , and N is the foot of the perpendicular from P on the transverse axis. The tantent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT.ON is equal to