Question

Show that all the chords of the curve ${3x}^2-y^2-2x+4y=0$ which subtend a right angle at the origin ?

Answer 1

Let (x,y) be a common point of chord y=mx+c, y=kx & 3x²−y²−2x+4y=0
First two give x & y which can be used in third to give (3−k²)c + (4k−2)(k−m) = 0
As a quadratic in k this is k²(4−c)−2k(2m+1)+(2m+3c) = 0
If the two values of k arising from this give perp lines thro O then $\frac{(2m+3c)}{(4-c)}=-1$
→ 2m+3c = −4+c → c=−m−2
Chord is thus y = mx–m−2 → y+2 = m(x−1) which always contains (1,−2)