locus of feet of perpendicular drawn from vertex of the parabola y2=4ax upon all such chords of the parabola which - Maths - Conic Sections

Question

locus of feet of perpendicular drawn from vertex of the parabola
y2=4ax upon all such chords of the parabola which subtend a right
angle at the vertex

Rahul Raj · Accepted Answer

dear student

let the points where line y=mx+c cut parabola beA(x1,y1) and B(x2,y2)y2=4ax(mx+c)2=4axm2x2+x(2mc-4a)x+c2=0x1x2=c2m2and y1y2=4acmas OA and OB are perpendicular
 O is originy1y2x1x2=-1c2m24acm=-1c=-4amso line is y=mx-4amlet the foot of perpendicular from origin on this line be (h,k)slope of OC=-1slope of line=-1mkh=-1mm=-hkalso (h,k) lie on y=mx-4amk=mh-4amk=-hkh-4a(-hk)h2+k2-4ah=0locus is x2+y2-4ax=0

regards

locus of feet of perpendicular drawn from vertex of the parabola y2=4ax upon all such chords of the parabola which subtend a right angle at the vertex