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# Solution for Find a Point on the Parabola Y = (X − 4)2, Where the Tangent is Parallel to the Chord Joining (4, 0) and (5, 1) ? - CBSE (Commerce) Class 12 - Mathematics

ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### Question

Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?

#### Solution

​Let: $f\left( x \right) = \left( x - 4 \right)^2 = x^2 - 8x + 16$

The tangent to the curve is parallel to the chord joining the points $\left( 4, 0 \right)$ and $\left( 5, 1 \right)$ .

Assume that the chord joins the points

$\left( a, f\left( a \right) \right)$ and $\left( b, f\left( b \right) \right)$ .
$\therefore$ $a = 4, b = 5$
The polynomial function is everywhere continuous and differentiable.
So,
$x^2 - 8x + 16$ is continuous on $\left[ 4, 5 \right]$ and differentiable on $\left( 4, 5 \right)$ .
Thus, both the conditions of Lagrange's theorem are satisfied.
Consequently, there exists $c \in \left( 4, 5 \right)$ such that
$f'\left( c \right) = \frac{f\left( 5 \right) - f\left( 4 \right)}{5 - 4}$ .
Now,
$f\left( x \right) = x^2 - 8x + 16 \Rightarrow$
$f'\left( x \right) = 2x - 8$,
$f\left( 5 \right) = 1, f\left( 4 \right) = 0$
$\therefore$  $f'\left( x \right) = \frac{f\left( 5 \right) - f\left( 4 \right)}{5 - 4}$
$\Rightarrow$ $2x - 8 = \frac{1}{1} \Rightarrow 2x = 9 \Rightarrow x = \frac{9}{2}$
Thus, $c = \frac{9}{2} \in \left( 4, 5 \right)$ such that ​
$f'\left( c \right) = \frac{f\left( 5 \right) - f\left( 4 \right)}{5 - 4}$ .
Clearly,
$f\left( c \right) = \left( \frac{9}{2} - 4 \right)^2 = \frac{1}{4}$
Thus,
$\left( c, f\left( c \right) \right)$ i.e.​
$\left( \frac{9}{2}, \frac{1}{4} \right)$,  is a point on the given curve where the tangent is parallel to the chord joining the points (4, 0) and (5, 1).
Is there an error in this question or solution?

#### APPEARS IN

Solution Find a Point on the Parabola Y = (X − 4)2, Where the Tangent is Parallel to the Chord Joining (4, 0) and (5, 1) ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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