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Question

Which of the following pairs is/are orthogonal?
(c and k are arbitrary constants.)

A
16x2+y2=c and y16=kx
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B
y=x+cex and x+2=y+key
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C
y=cx2 and x2+2y2=k
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D
x2y2=c and xy=k
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Solution

The correct options are
A 16x2+y2=c and y16=kx
B y=x+cex and x+2=y+key
C y=cx2 and x2+2y2=k
D x2y2=c and xy=k
16x2+y2=c
Differentiating w.r.t. x
32x+2ydydx=0
dydx=m1=16xy
and y16=kx
Differentiating w.r.t. x
16y15dydx=k
dydx=m2=k16y15
m1m2=16xy×k16y15=xy16×k
m1m2=xy16×y16x=1

y=x+cex
dydx=1cex=1(yx)=(yx1)
dydx=m1=(yx1)
and x+2=y+key
dydxkdydxey=1
dydx[1key]=1
dydx=m2=11key=1yx1
m1m2=1

y=cx2
dydx=2cx=2xyx2=2yx
dydx=m1=2yx
and x2+2y2=k
2x+4ydydx=0
dydx=m2=x2y
m1m2=1

x2y2=c
2x2ydydx=0
dydx=m1=xy
and xy=k
xdydx+y=0
dydx=m2=yx
m1m2=1

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