Question

Geometry

tangent normal 5. Let P be a point on the ellipse with equation +5 = 1, where a >b>0, b2 = a (1 - e?), and 0 <e<1. (a) If P has coordinates (a cost, b sin :), determine the equation of the normal at P to the ellipse. (b) Determine the coordinates of the point where the normal in part (a) meets the axis y = 0. (c) Let F be the focus with coordinates (a,0). Prove that QF = e. PF. 4 3 1

Answer 1

as we have the given allipse. a>b> let (2, 3) be any point as the ellipse.. Then the equation of the targest to the ellopse m e at the point (x, ) is W tymi a The scope of the tangent is f-ber Therefore the slope of the normal will be am bra Then (c) the equation of the normal at the point -3, = A (2-04) - 174 F xular or x-u - Bor 3/62 my and bon carb? Now Abe Here (xs, W) is an arbitrary point Now the equation of the normal to the elipse at Bra at the poolest (acost, brint) is geren, by are born - az.br a cost 6 sont or art cost 6 cent This is the required gorgnal at normal at pa

Now the coordinate of the point & where the normal in part (a) meets the caseis voor Now we put z=0 in equation 1 we get 09x a an - 62 cost Q & is or & - au-brcost Thus the coordinate of the point given by fa- breast, o)