Tangent to hyperbola xy = c ^2 at point P intersects the x -axis at T and the y -axis at T ^ . Normal hyperbola at P intersects the x -axis at N and the y -axis at N ^ . If the area of the triangles PNT a PN'T' are and ^ respectively then 1 + 1 ^ , is equal to

Step-by-step solution

Equation of tangent x+t^2 y=2 c t T (2 c t, 0) and T ^ (0, 2 c t ) Equation of normal t ^3 x - ty = c ( t ^4-1 ) ; N [ c ( t - 1 t ^3 ), 0 ] ; N [0, c ( 1 t - t ^3 ) ] aligned & N T= c t ^3 + ct ; N ^ T ^ = ct ^3+ c t & = 1 2 | c t ^3 + ct | c t 1 = 2 t ^4 c ^2 (1+ t ^4 ) and ^ = 1 2 | ct ^3+ c t | ct & 1 ^ = 2 c ^2 (1+ t ^4 ) 1 + 1 ^ = 2 c ^2 aligned