Answer to Question #236490 in Geometry for Mohamed ali

Question

1. Find the equation of the locus of the mid-point of the chords of the curve 5x^2 + 4y^ 2 = 6 with gradient 3.
2. P is a variable point on the curve x^ 2 + 4y^ 2 = 25 and A is the point (2, 6) . Find the equation of the locus of the mid-point of AP.

Answer 1

Question 1

Let (h,k) is the mid-point of chords

T= S1

T=5hx+4ky-6

S1= 5h²+4k² -6

-5h/4k=3

-5h-12k=0

Hence the locus is:

5x+12y=0

Question 2

Let M(h,k) is the mid-point of AP

A(2,6), P(x,y)

(x+2)/2= h x=2h-2

(y+6)/2=k y=2k-6

(2h-2)²+4(2k-6)² =25

4h² +16k² -8h-96k+123=0

Hence locus is

4x² +16y²-8x-96y+123=0