The normal to the curve at #y=asqrtx+b/sqrtx# where a and b are constants is #4x+y=22# at the point where x=4 and how do you find a and b?

Answer 1

#a=2, and, b=4#.

We know that the slope of normal to the curve

# : y = f(x) "at" x=4 "is" -1/(f'(4))#.
We have, #y=f(x)=asqrtx+b/sqrtx#
#rArr dy/dx=f'(x)=a/(2sqrtx)+b(-1/2*x^(-3/2))#
#rArr f'(4)=a/4-b/16=(4a-b)/16#
#:. "the slope of normal at" #x=4# "is" -1/(f'(4))=16/(b-4a)#
But, from the eqn. of normal, the slope is #-4#. hence,
#16/(b-4a)=-4 rArr 4a-b=4..............................................(1)#
When #x=4, y=f(4)=2a+b/2.#
Hence, the pt. of contact is #(4,2a+b/2)#, which lies on the normal
line # : 4x+y=22#, so that, we get,
#16+2a+b/2=22, i.e., 2a+b/2=6, or, 4a+b=12......(2)#.
Solving #(1) & (2), a=2, and, b=4#.

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Answer 2

To find the values of a and b, we need to use the given information about the normal to the curve.

First, let's find the derivative of the curve equation y = a√x + b/√x with respect to x.

dy/dx = (1/2) * a * x^(-1/2) - (1/2) * b * x^(-3/2)

Next, we know that the normal to the curve is perpendicular to the tangent at the given point (x=4).

The slope of the tangent at x=4 can be found by substituting x=4 into the derivative:

dy/dx = (1/2) * a * 4^(-1/2) - (1/2) * b * 4^(-3/2)

Now, we can determine the slope of the normal by taking the negative reciprocal of the tangent's slope:

m_normal = -1 / [(1/2) * a * 4^(-1/2) - (1/2) * b * 4^(-3/2)]

Since the normal passes through the point (4, y), we can substitute these values into the equation of the normal line (4x + y = 22) to find the value of y:

4 * 4 + y = 22

Solving this equation will give us the value of y.

Finally, we can substitute the values of x=4 and y into the derivative equation and solve for a and b.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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