Q. 55.0( 3 Votes )
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8.
Answer :
Given:
Curves 2x = y2 ...(1)
& 2xy = k ...(2)
We have to prove that two curves cut at right angles if k2 = 8
Now ,Differentiating curves (1) & (2) w.r.t x, we get
⇒ 2x = y2
⇒ 2 = 2y.
⇒ 
m1
...(3)
⇒ 2xy = k
Differentiating above w.r.t x,
⇒ 2(1×
) = 0
⇒ 
= 0
m2
...(4)
Since m1 and m2 cuts orthogonally,
⇒ 
×
1
⇒ 
1
⇒ x = 1
Now , Solving (1) & (2),we get,
2xy = k & 2x = y2
⇒ (y2)y = k
⇒ y3 = k
⇒ y
Substituting y
in 2x = y2,we get,
⇒ 2x = (
)2
⇒ 2×1
⇒ 
2
⇒ 
23
⇒ 
8
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