find the value of P for when the curves X2=9p(9-y) and X2=p(y+1) cut each other at right angle - Maths - Continuity and Differentiability

Question

find the value of P for when the curves X2=9p(9-y) and
X2=p(y+1) cut each other at right angle

Ajanta Trivedi · Accepted Answer

the given curves are :
x2=9p(9-y)  (1) and x2=p(y+1) 
(2)let us find the intersection point of curve (1) and curve (2):9p(9-y)=p(y+1)81p-9py=py+p80p-10py=0p(8-y)=0since p≠0 therefore y=8 
(2)now we will find the slope of the given curve:differentiate eq(1), we have:2x=-9p
dydx⇒2x=-9p
m1  [where m1 be the slope of curve (1) at any pointm1=-29
xp
differentiate eq(2), we have:
2x=p dydx⇒2x=p
m2 [where m2 be the slope of the curve (2) at any pointm2=2xpsince both the curve cut each other at right angle,m1
m2=-1-29 xp 2xp=-149 x2p2=1⇒x2=94
p2substituting x2 in terms of p and y=8 in eq(1):
94 p2=9p(9-8)14
p2=pp2-4p=0p(p-4)=0since p≠0 therefore p=4
hope this helps you