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Maths-

General

Easy

Question

If the tangent at the point $open parentheses a t to the power of 2 end exponent comma a t to the power of 3 end exponent close parentheses$ on the curve $a y to the power of 2 end exponent equals x to the power of 3 end exponent$ meets the curve again at

$(\frac{a t^{2}}{4}, - \frac{a t^{3}}{8})$
$(\frac{a t^{2}}{4}, 8 at)$
$(\frac{a t^{2}}{2}, 2 a t^{2})$
$(a t^{2}, 2 a t)$

hint Hint:

We are give the point of tangency. It is (at², at³). The given curve is. We have to find the point where tangent intersects the curve again. We will find the slope using the curve. Then, we will use two point form to find the point.

The correct answer is: $open parentheses fraction numerator a t squared over denominator 4 end fraction comma negative fraction numerator a t cubed over denominator 8 end fraction close parentheses$

The given curve is ay² = x³
The point of tangency is (at², at³)
Let the point that tangent intersects again be (as², as³)
We can find the slope of tangent by differentiating the given curve w.r.t x

$2 a y fraction numerator d y over denominator d x end fraction equals 3 x squared space space space space space fraction numerator d y over denominator d x end fraction equals fraction numerator 3 x squared over denominator 2 a y end fraction T h e space s l o p e space i s space m space equals space open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis a t squared comma space a t cubed right parenthesis end subscript space space space space space space equals fraction numerator 3 left parenthesis a t squared right parenthesis squared over denominator 2 a left parenthesis a t cubed right parenthesis end fraction space space space space space space equals fraction numerator 3 a squared t to the power of 4 over denominator 2 a squared t cubed end fraction space space space m space space equals fraction numerator 3 t over denominator 2 end fraction$
Now, we will use two point form to solve the question.
If (x₁, y₁) and (x₂, y₂) be two points on the line then the slope is
$m equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction$
For this question, it will be
$m equals fraction numerator a s cubed minus a t cubed over denominator a s squared minus a t squared end fraction$
We will solve it further and find the value.
$3 over 2 t equals fraction numerator a open parentheses s cubed minus t cubed close parentheses over denominator a open parentheses s squared minus t squared close parentheses end fraction space space space space space space space equals fraction numerator left parenthesis s space minus space t right parenthesis left parenthesis s squared plus s t space plus t squared right parenthesis over denominator left parenthesis s space minus t right parenthesis left parenthesis s space plus t right parenthesis end fraction space space space space space space space space space space space space space space space space space space... left curly bracket space left parenthesis a cubed space minus space b cubed space equals left parenthesis a space minus space b right parenthesis left parenthesis a squared space plus space a b space plus space b squared right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space a squared space minus space b squared space equals space left parenthesis a space minus space b right parenthesis left parenthesis a space plus space b right parenthesis space space space space right curly bracket$
$3 over 2 t equals fraction numerator s squared plus s t plus t squared over denominator s plus t end fraction 3 t left parenthesis s space plus space t right parenthesis space equals space 2 left parenthesis s squared space plus space s t space plus space t squared right parenthesis space space 3 s t space plus 3 t squared space equals 2 s squared plus 2 s t space plus space 2 t squared space space space space space space space space space space space space space 0 space equals space 2 s squared plus 2 s t space minus 3 s t space plus 2 t squared minus 3 t squared 2 s squared minus s t space minus t squared space equals space 0 W e space w i l l space u s e space f a c t o r i z a t i o n space m e t h o d. 2 s squared minus 2 s t space plus s t space minus t squared space equals space 0 2 s left parenthesis s space minus space t right parenthesis space plus space t left parenthesis s space minus t space right parenthesis space equals space 0 left parenthesis 2 s space plus space t right parenthesis left parenthesis s space minus space t right parenthesis space equals space 0$
(s - t) = 0
s = t
It means the same point of tangency. We want to find different point.
(2s + t) = 0
s = $negative t over 2$
So, the point will be
$open parentheses a s squared comma a s cubed close parentheses equals open parentheses a open parentheses negative t over 2 close parentheses squared comma a open parentheses fraction numerator negative t over denominator 2 end fraction close parentheses cubed close parentheses space space space space space space space space space space space space space space space space space equals open parentheses fraction numerator a t squared over denominator 4 end fraction comma space fraction numerator negative a t cubed over denominator 8 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space$
This is the required answer.

For such questions, we should know the formulas to find slope.

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