Question

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Answer 1

Given:

Gradient to the curve xy+ax+by=0 at (1,1)= 2

To Find:

• Value of a+2b

Solution:

We know that,

↬ Gradient means slope or differentiation

↬ $\bf{\dfrac{d}{dx}(a\times b)=a \dfrac{d}{dx}(b)+b\dfrac{d}{dx}(a)}$

where a and b are functions of x

↬ $\bf{\dfrac{d}{dx}(x)=1}$

So,

It is given that $\sf{\dfrac{dy}{dx}=2}$

Now,

On differentiating given curve w.r.t x, we get

$\longrightarrow\sf{x\dfrac{dy}{dx}+y(1)+a(1)+b\dfrac{dy}{dx}=0}$

$\longrightarrow\sf{x\dfrac{dy}{dx}+y+a+b\dfrac{dy}{dx}=0}$

$\longrightarrow\sf{x\dfrac{dy}{dx}+b\dfrac{dy}{dx}+y+a=0}$

$\longrightarrow\sf{(x+b)\dfrac{dy}{dx}+y+a=0}$

Now, above curve passes through (1,1), so it will satisfy the equation

$\longrightarrow\sf{(1+b)\dfrac{dy}{dx}+1+a=0}$

Also, after putting the value of dy/dx, we get

$\longrightarrow\sf{(1+b)(2)+1+a=0}$

$\longrightarrow\sf{2+2b+1+a=0}$

$\longrightarrow\sf{a+2b+3=0}$

$\longrightarrow\sf\pink{a+2b=-3}$

Hence, the value of a+2b is -3.