Question

if the curves y2=4ax and xy=c2 cut at right angles ,prove that

C4=32a4

Answer 1

curves y² = 4ax and xy = c² cut at right angles .
means, at cutting point , slope of tangent of curve= 4ax} × slope of tangent of curve {xy = c²} = -1

slope of tangent of curve y² = 4ax :
differentiate with respect to x ,
2ydy/dx = 4a
dy/dx = m = 2a/y .......... (i)

slope of tangent of curve xy = c² :
differentiate with respect to x ,
xdy + ydx = 0
dy/dx =m' = -y/x .........(ii)

Let curves cut at (p,q)
then, y² = 4ax at (p,q) is satisfied so, q² = 4ap
and xy = c² at (p,q) is satisfied so, pq = c²
{c²/p}²= 4ap
c⁴ = 4ap³ ...........(iii)
and put (p,q) in slope too,
then, slope of tangent of curve y² = 4ax
dy/dx = m = 2a/q
and slope of tangent of curve xy = c²
dy/dx = m' = -q/p

now, 2a/q × -q/p = -1
2a/p = 1
2a = p .......(iv)

from equation (III) and (iv) ,
c⁴ = 4ap³ = 4a(2a)³ = 32a⁴
c⁴ = 32a⁴ , hence proved

Answer 2

Answer:

Secondary SchoolMath 13+7 pts

If the curves y2=4ax and xy=c2 cut at right angles ,prove that

C4=32a4

Report by NAGASAKI6480 28.08.2018

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abhi178

Abhi178 The Sage

curves y² = 4ax and xy = c² cut at right angles .

means, at cutting point , slope of tangent of curve= 4ax} × slope of tangent of curve {xy = c²} = -1

slope of tangent of curve y² = 4ax :

differentiate with respect to x ,

2ydy/dx = 4a

dy/dx = m = 2a/y .......... (i)

slope of tangent of curve xy = c² :

differentiate with respect to x ,

xdy + ydx = 0

dy/dx =m' = -y/x .........(ii)

Let curves cut at (p,q)

then, y² = 4ax at (p,q) is satisfied so, q² = 4ap

and xy = c² at (p,q) is satisfied so, pq = c²

{c²/p}²= 4ap

c⁴ = 4ap³ ...........(iii)

and put (p,q) in slope too,

then, slope of tangent of curve y² = 4ax

dy/dx = m = 2a/q

and slope of tangent of curve xy = c²

dy/dx = m' = -q/p

now, 2a/q × -q/p = -1

2a/p = 1

2a = p .......(iv)

from equation (III) and (iv) ,

c⁴ = 4ap³ = 4a(2a)³ = 32a⁴

c⁴ = 32a⁴ , hence proved

Step-by-step explanation: