A normal to the hperbola `(X^(2))/(a^(2))-(y^(2))/(b^(2))=1` meets the axes at 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)`.

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)dot

The line y= mx +c touches the hyperbola b^(2) x^(2) - a^(2) y^(2) = a^(2)b^(2) if-

A normal to the hyperbola, 4x^2_9y^2 = 36 meets the co-ordinate axes x and y at A and B. respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is :-

Ifthe normal at P to the rectangular hyperbola x^2-y^2=4 meets the axes in G and g and C is the centre of the hyperbola, then

A tangent to the hyperbola x^(2)-2y^(2)=4 meets x-axis at P and y-aixs at Q. Lines PR and QR are drawn such that OPRQ is a rectangle (where O is origin).Find the locus of R.

If the normal at a point P to the hyperbola x^2/a^2 - y^2/b^2 =1 meets the x-axis at G , show that the SG = eSP.S being the focus of the hyperbola.

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.

For 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.

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

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 tangent 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