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
B
1
C
1
D
2

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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