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Maths-

General

Easy

Question

The length of the normal to the curve $y equals a c o s space h open parentheses x over a close parentheses$ at any point varies as

Ordinate
Abscissa
Square of the abscissa
Square of the ordinate

hint Hint:

We are given a curve. We have to tell how the length of the normal varies. We will use the formula of length to normal to find the answer.

The correct answer is: Square of the ordinate

The given curve is $y equals a cos h open parentheses x over a close parentheses$
Ordinate means the y-axis
And abscissa means x-axis
The formula of length of the normal at point (x,y) is given by $L e n g t h space equals space open vertical bar y subscript 1 square root of 1 space plus space m end root close vertical bar$

Here, m = $open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis x subscript 1 comma y subscript 1 right parenthesis end subscript$
Now, we will find the value of $fraction numerator d y over denominator d x end fraction$
$y equals a cos h open parentheses x over a close parentheses fraction numerator d y over denominator d x end fraction equals a sin h open parentheses x over a close parentheses fraction numerator d over denominator d x end fraction open parentheses x over a close parentheses space space space space space space space space space space space space space space space space space... left curly bracket fraction numerator d over denominator d x end fraction cos h x space equals space minus sin h x right curly bracket space space space space space space space equals space minus a sin h open parentheses x over a close parentheses space cross times 1 over a space space space space space space space equals negative sin h open parentheses x over a close parentheses$
$m space equals open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis x subscript 1 comma y subscript 1 right parenthesis end subscript space space space space equals negative sin h open parentheses x subscript 1 over a close parentheses$
We will substitute this value in the formula of length.
$L e n g t h equals vertical line y subscript 1 square root of 1 plus m squared end root vertical line space space space space space space space space space space space space space equals vertical line y subscript 1 square root of 1 space plus space left parenthesis negative sin h x over a end root right parenthesis squared vertical line space space space space space space space space space space space space space equals vertical line y subscript 1 square root of cos h squared x over a end root vertical line space space space space space space space space space space space space space space... left curly bracket 1 space plus space sin h squared x space equals space cos h squared x right parenthesis space space space space space space space space space space space space space equals open vertical bar y subscript 1 cos h x over a close vertical bar space space y subscript 1 space end subscript equals space a cos h x over a s o comma space cos h x over a equals y subscript 1 over a L e n g t h space equals space vertical line y subscript 1 cross times y subscript 1 over a vertical line space space space space space space space space space space space space space space equals y subscript 1 squared over a$
Length is proportional to y₁² as "a" is constant.
We can see the length of the normal depends on the square of y. It means the length of normal varies square of ordinate.

For such questions, we should know the formula of length of normal.

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