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+1 vote
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in Calculus by (115k points)
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is:
1. 1
2. 0
3. -6
4. 6

1 Answer

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Best answer
Correct Answer - Option 4 : 6

Concept:

Intersecting curves are orthogonal if they meet at right angles.

If the curves are orthogonal, then the product of the slopes of the tangents (or normals) at the point of intersection is equal to -1.

The slope of the tangent to the curve y = f(x) at a point (a, b) is given by m = \(\rm \left(\frac{dy}{dx}\right)_{(a,b)}\).

 

Calculation:

Given curve are ay + x2 = 7 and x3 = y

ay + x2 = 7

Differentiation with respect to x, we get

⇒ a\(\rm \frac {dy}{dx}\) + 2x = 0

⇒ \(\rm \frac {dy}{dx} = \frac {-2x}a\) = Slope of 1st curve

Now, x3 = y

Differentiation with respect to x, we get

⇒ 3x\(\rm \frac {dy}{dx}\) = Slope of 2nd curve

It is given, curve cut orthogonally at (1, 1)

So, The product of the slopes of the tangents to both the curves at (1, 1) must be -1.

⇒ \(\rm \left(\frac{-2x}{a}\right)_{(1,1)}\times\left(3x^2\right)_{(1,1)}=-1\)

⇒ a = 6.

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