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Question
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is ______.
Options
1
0
– 6
6
Solution
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is 6.
Explanation:
Equation of the given curves are ay + x2 = 7 .....(i)
And x3 = y .....(ii)
Differentiating eq. (i) w.r.t. x, we have
`"a" "dy"/"dx" + 2x` = 0
⇒ `"dy"/"dx" = - (2x)/"a"`
∴ m1 = `- (2x)/"a"` ......`("m"_1 = "dy"/"dx")`
Now differentiating eq. (ii) w.r.t. x, we get
3x2 = `"dy"/"dx"`
⇒ m2 = `3x^2` .....`("m"_2 = "dy"/"dx")`
The two curves are said to be orthogonal if the angle between the tangents at the point of intersection is 90°.
∴ m1 × m2 = – 1
⇒ `(-2x)/"a" xx 3x^2` = – 1
⇒ `(-6x^3)/"a"` = – 1
⇒ 6x3 = a
(1, 1) is the point of intersection of two curves.
∴ 6(1)3 = a
So a = 6
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