Question

What is the equation of the family of curves orthogonal to the family y=ax3
(A) X2+3y2=a2
(B) x2+3y2=2
(C) both A and B true
(D) none of these​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

The equation of the family of curves orthogonal to the family

$\sf{y = a {x}^{3} }$

(A) x² + 3y² = a²

(B) x² + 3y² = 2

(C) both A and B true

(D) none of these

EVALUATION

Here the given equation of the family of curves is

$\sf{y = a {x}^{3} } \: \: \: \: - - - - (1)$

Differentiating both sides with respect to x we get

$\displaystyle \sf{ \frac{dy}{dx} = 3a {x}^{2} }$

From Equation 1 Putting the value of a we get

$\displaystyle \sf{ \frac{dy}{dx} = 3 \times \frac{y}{ {x}^{3} } \times {x}^{2} }$

$\displaystyle \sf{ \implies \frac{dy}{dx} = \frac{3y}{x} }$

Which is the differential equation of the given family of curves

Now the differential equation of the family of curves orthogonal to the given family is

$\displaystyle \sf{ - \frac{dx}{dy} = \frac{3y}{x} }$

$\displaystyle \sf{ \implies x \: dx = - 3y \: dy }$

On integration we

$\displaystyle \sf{ \int x \: dx = - 3 \int \: y \: dy }$

$\displaystyle \sf{ \implies \int x \: dx + 3 \int \: y \: dy = 0 }$

$\displaystyle \sf{ \implies \: \frac{ {x}^{2} }{2} + \frac{3 {y}^{2} }{2} = \frac{ {a}^{2} }{2} }$

Where a is constant

$\displaystyle \sf{ \implies \: {x}^{2} + 3 {y}^{2} = {a}^{2} }$

Which is the required equation of the family of curves orthogonal to the given family of curves

FINAL ANSWER

Hence the correct option is

(A) x² + 3y² = a²

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. Solve the differential equation :-

(cosx.cosy - cotx) dx - (sinx.siny) dy=0

https://brainly.in/question/23945760

2. Find the complete integral of p-q=0

https://brainly.in/question/23947735