Ellipse

The normal is
4x sec ϕ - 2 y cosec ϕ = 12
Now, the points Q and M are given by
Q = (3 cos ϕ , 0)
M = (α , β)
Therefore, α =

3cosϕ+4cosϕ

2

=

7

2

cosϕ ⇒ cos ϕ =

2

7

α
and β = sin ϕ ; cos2ϕ+sin2ϕ = 1.
Therefore,

4

49

α2+β2 = 1 ⇒

4

49

x2+y2 = 1
Hence, the rectum is x = ± 2√3
Hence,

48

49

+y2 = 1 ⇒ y = ±

1

7

(±2√3,±

1

7

)
Hence, the locus of M intersects the latus rectum of the given ellipse at the points