Maths-
General
Easy
Question
The tangent at (
,
–
) on the curve y =
–
meets the curve again at Q, then abscissa of Q must be -
- 1 + 2t
- 1 – 2t
- –1 –2t
- 2t – 1
Hint:
Slope of tangent = ![fraction numerator d y over denominator d x end fraction](data:image/png;base64,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)
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 =
–
and Point P(
,
–
)
![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](data:image/png;base64,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)
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](data:image/png;base64,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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](data:image/png;base64,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)
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](data:image/png;base64,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