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Question

If the line $$y=4x-5$$ touches to the curve $${ y }^{ 2 }=a{ x }^{ 3 }+b$$ at the point $$(2,3)$$ then $$7a+2b=$$


A
0
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B
1
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C
1
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D
2
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Solution

The correct option is B $$0$$
The slope of given line $$y=4x-5$$ is  $$m=4$$.

Now, since the given line touches the curve at $$(2,3)$$, the slope of curve $$\dfrac{dy}{dx}$$ at the given point is equal to the slope $$m$$ of  the line.

Hence, for slope of curve, differentiating both sides w.r.t x:-

$$2y\dfrac{dy}{dx}=3ax^2$$

Putting the value $$x=2 $$ and $$y=3$$

$$6\dfrac{dy}{dx}=3a\times 4 \implies 6\times4=12a$$

 $$\implies a=2$$

Now, the given point lies on the curve, so it must satisfy the equation of the curve:-

$$\implies (3)^2=2\times (2)^3+b$$

$$\implies b=-7$$

$$7a+2b=(7\times 2-2\times 7)=0$$

Hence, answer is option-(A).

Mathematics

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