a 1 is constant Find the constant also a b2 2 16 Find the

a, + = 1 is constant. Find the constant also. a b2 2 (16) Find the locus of the foot of the perpendicular from the centre of the elipse T + 22-1 on the chord joining two points whose eccentric angles differ by 2

Solution

Let point P be a point with eccentric angle and let Q be a point with eccentric angle ( - /2) as per the hypothesis on any chord

Hence the coordinates are

P = (acos, bsin)
Q = [acos( - /2), bsin( - /2)]
Equivalently, Q = (asin, -bcos)

Let M be the midpoint of chord PQ. Point M has these coordinates:

M = [(a/2)(sin + cos), (b/2)(sin - cos)]

Let (x, y) be the coordinates of the foot of perpendicular from the centre of the given ellipse on this chord PQ, i.e., the locus

x = (a/2)(sin + cos)
x² = (a/2)²(sin² + 2sincos + cos²)
x² = (a/2)²(1 + 2sincos)

y = (b/2)(sin - cos)
y² = (b/2)²(sin² - 2sincos + cos²)
y² = (b/2)²(1 - 2sincos)

x² + y² = (a/2)²(1 + 2sincos) + (b/2)²(1 - 2sincos)
x²/(a/2)² + y²/(b/2)² = (1 + 2sincos) + (1 - 2sincos)
x²/(a/2)² + y²/(b/2)² = 2
x²/[a/(2)]² + y²/[b/(2)]² = 1