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Question

If the tangent at ‘t’ on the curve y = $8 t cubed space minus 1$ , x = $4 t squared space plus space 3$ meets the curve again at $t subscript 1$ and is normal to the curve at that point, then value of t must be -

± $\frac{1}{3}$
± $\frac{1}{\sqrt{2}}$
± $\frac{\sqrt{2}}{3}$
None of these

hint Hint:

Slope of tangent = $fraction numerator d y over denominator d x end fraction space equals space fraction numerator begin display style bevelled fraction numerator d y over denominator d t end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d t end fraction end style end fraction$

The correct answer is: ± $1 third$

Slope of tangent = $fraction numerator d y over denominator d x end fraction space equals space fraction numerator begin display style bevelled fraction numerator d y over denominator d t end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d t end fraction end style end fraction$
Given : y = $8 t cubed space minus 1$ and x = $4 t squared space plus space 3$
$fraction numerator d y over denominator d t end fraction space equals space 24 t squared space a n d space fraction numerator d x over denominator d t end fraction space equals space 8 t$
$fraction numerator d y over denominator d x end fraction space equals space fraction numerator begin display style bevelled fraction numerator d y over denominator d t end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d t end fraction end style end fraction$ = $fraction numerator 24 t squared over denominator 8 t end fraction space equals space 3 t$
Equation tangent at point 't'
y -y1 = m(x -x1)
$rightwards double arrow y space minus space 8 t cubed space plus space 1 space equals space space 3 t left parenthesis space x minus space 4 t squared space minus 3 right parenthesis A t space p o i n t space t subscript 1 rightwards double arrow 8 t subscript 1 cubed minus 1 space minus space 8 left parenthesis t right parenthesis cubed space plus space 1 space equals space 3 t left parenthesis 4 t subscript 1 squared space plus space 3 space minus space 4 left parenthesis t right parenthesis squared space minus 3 right parenthesis rightwards double arrow 8 left parenthesis t subscript 1 cubed space minus space t cubed right parenthesis space equals space 12 t space left parenthesis t subscript 1 squared space minus space t squared right parenthesis rightwards double arrow 8 left parenthesis t subscript 1 space minus space t right parenthesis left parenthesis t subscript 1 squared space plus space t squared space plus t t subscript 1 right parenthesis space equals space 12 t space left parenthesis space t subscript 1 space minus t right parenthesis left parenthesis t subscript 1 space plus space t right parenthesis rightwards double arrow 2 t subscript 1 squared space space plus space 2 t squared space plus 2 t subscript 1 t space equals space 3 t t subscript 1 space plus space 3 t squared rightwards double arrow 2 t subscript 1 squared space minus space t t subscript 1 space minus space t squared space equals space 0 rightwards double arrow 2 t subscript 1 squared space minus 2 space t t subscript 1 space plus space t t subscript 1 space minus space t squared space equals space 0 rightwards double arrow 2 t subscript 1 left parenthesis t subscript 1 space minus t right parenthesis space plus space t left parenthesis t subscript 1 space minus t right parenthesis equals space 0 rightwards double arrow left parenthesis 2 t subscript 1 space end subscript plus space t right parenthesis left parenthesis t subscript 1 space minus t right parenthesis space equals space 0 rightwards double arrow t space equals space t subscript 1 left parenthesis n o t space p o s s i b l e right parenthesis space. space t space equals space minus 2 t subscript 1 space end subscript rightwards double arrow T h u s comma space t space equals space minus 2 t subscript 1 space end subscript rightwards double arrow t subscript 1 space end subscript space equals negative space t over 2 rightwards double arrow S l o p e space o f space n o r m a l space equals space minus fraction numerator 1 over denominator 3 t end fraction$
Equation of normal
$y space minus space 8 t cubed space plus 1 space equals space minus fraction numerator 1 over denominator 3 t end fraction left parenthesis x space minus space 4 t squared space minus 3 right parenthesis A t space p o i n t space t subscript 1 rightwards double arrow 8 t subscript 1 cubed minus 1 space minus space 8 left parenthesis t right parenthesis cubed space plus space 1 space equals space space minus fraction numerator 1 over denominator 3 t end fraction left parenthesis 4 t subscript 1 squared space plus space 3 space minus space 4 left parenthesis t right parenthesis squared space minus 3 right parenthesis rightwards double arrow 8 left parenthesis t subscript 1 to the power of 3 space end exponent space minus space t cubed right parenthesis space equals space minus fraction numerator 4 over denominator 3 t end fraction space left parenthesis space t subscript 1 squared space space minus space t squared right parenthesis$

$rightwards double arrow 6 t space left parenthesis space t subscript 1 cubed space minus t cubed right parenthesis space equals space left parenthesis t squared space minus space t subscript 1 squared right parenthesis rightwards double arrow negative space 6 t space space left parenthesis space t squared space plus space t subscript 1 squared space plus t t subscript 1 right parenthesis space equals space t space plus space t subscript 1 S u b s t i t u t e space t h e space v a l u e space o f space t subscript 1 rightwards double arrow negative space 6 t space space left parenthesis space t squared space plus space t over 4 squared space minus space t t over 2 right parenthesis space equals space t space minus t over 2$
$fraction numerator rightwards double arrow 6 over denominator 4 end fraction left parenthesis 5 t squared space minus space 2 t squared right parenthesis space equals space 1 half rightwards double arrow t squared space equals space 1 over 9 rightwards double arrow t space equals space plus-or-minus 1 third$

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