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Maths-

General

Easy

Question

The tangent at ( $t$ , $t squared$ – $t cubed$ ) on the curve y = $x squared$ – $x cubed$ meets the curve again at Q, then abscissa of Q must be -

1 + 2t
1 – 2t
–1 –2t
2t – 1

hint Hint:

Slope of tangent = $fraction numerator d y over denominator d x end fraction$
find equation of tangent at P, then find the intersection point of the tangent and curve by substituting the given values in the option, and whosoever satisfies the equation is the required point.

The correct answer is: 1 – 2t

Given : y = $x squared$ – $x cubed$ and Point P( $t$ , $t squared$ – $t cubed$ )
$S l o p e space equals fraction numerator d y over denominator d x end fraction space equals space 2 x space minus space 3 x squared A t space p o i n t space left parenthesis t comma space t squared space – space t cubed right parenthesis rightwards double arrow fraction numerator d y over denominator d x end fraction space equals space 2 t space minus space 3 t squared$
Equation of tangent at point P
$rightwards double arrow y space minus left parenthesis space t squared space minus t cubed right parenthesis space equals space left parenthesis 2 t space minus space 3 t squared right parenthesis left parenthesis x minus t right parenthesis rightwards double arrow y space minus space t squared space plus space t cubed space equals space x t left parenthesis 2 minus 3 t right parenthesis space minus 2 t squared space plus space 3 t cubed rightwards double arrow y space equals space x t left parenthesis 2 minus 3 t right parenthesis space minus space t squared space space plus space 2 t cubed$
Find intersection point of the tangent and the curve by equating values of y for curve and tangent
$rightwards double arrow x squared space minus x cubed space equals space x left parenthesis 2 t space minus 3 t squared right parenthesis space plus space 2 t cubed space minus t squared rightwards double arrow space space x cubed minus space x squared space plus space x left parenthesis 2 t space minus space 3 t squared right parenthesis space plus space t squared left parenthesis 2 t space minus space 1 right parenthesis space equals space 0 f o r space x space equals space 2 t plus 1 rightwards double arrow 8 t cubed space plus space 1 space plus space 12 t squared space plus space 6 t space minus space 4 t squared space minus 1 space minus 4 t space plus 4 t squared space minus 6 t cubed space plus 2 t space minus space 3 t squared space plus 2 t cubed space minus t squared space equals space d o e s n o t space s a t i s f y$

for x = 1-2t
$rightwards double arrow 1 minus space 8 t cubed space minus 6 t space plus 12 t squared space minus 1 minus 4 t squared space plus 4 t space plus 2 t minus 3 t squared space minus 4 t squared space plus 6 t cubed space plus 2 t cubed space minus t squared space equals space 0 T h u s space i t space s a t i s f i e s space left parenthesis 1 minus 2 t right parenthesis$

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