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Maths-

General

Easy

Question

The slope of the tangent to the curves $x equals t to the power of 2 end exponent plus 3 t minus 8 comma blank y equals 2 t to the power of 2 end exponent minus 2 t minus 5$ at the point ( $2 comma blank minus 1 right parenthesis blank$ is

22/7
6/7
-6
-7

The correct answer is: 6/7

Given curve are
$x equals t to the power of 2 end exponent minus 3 t minus 8 comma blank y equals 2 t to the power of 2 end exponent minus 2 t minus 5$ …(i)

When $x equals 2$ , then

$2 equals t to the power of 2 end exponent plus 3 t minus 8$

$rightwards double arrow blank t to the power of 2 end exponent plus 3 t minus 10 equals 0 blank rightwards double arrow blank t equals 2 comma blank minus 5$ ...(ii)

When $y equals negative 1$ , then

$negative 1 equals 2 t to the power of 2 end exponent minus 2 t minus 5$

$rightwards double arrow blank 2 t to the power of 2 end exponent minus 2 t minus 4 equals 0 blank rightwards double arrow blank t equals negative 1 comma blank 2$ ...(iii)

From Eqs. (ii) and (iii), $t equals 2$

On differentiating Eq. (i) w.r.t. $t$ respectively, we get

$fraction numerator d x over denominator d t end fraction equals 2 t plus 3$ and $fraction numerator d y over denominator d t end fraction equals 4 t minus 2$

$therefore blank fraction numerator d y over denominator d x end fraction equals fraction numerator d y divided by d t over denominator d x divided by d t end fraction equals fraction numerator 4 t minus 2 over denominator 2 t plus 3 end fraction$

$rightwards double arrow blank open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis t equals 2 right parenthesis end subscript equals fraction numerator 8 minus 2 over denominator 4 plus 3 end fraction equals fraction numerator 6 over denominator 7 end fraction$

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