Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

Find the points on the curve

x^{2} + y^{2} = 13

, the tangent at each one of which is parallel to

2 x + 3 y = 7

.

Accepted Answer

Let the points be $(x_{1}, y_{1})$

And slope of tangent at $(x_{1}, y_{1})$ be $m$ .

We know that if tangent at $(x_{1}, y_{1})$ is parallel to a line then its slope will be equal to slope of line.

And equation of line is,

$2 x + 3 y = 7$

$\Rightarrow y = \frac{- 2 x}{3} + \frac{7}{3}$

So, slope of line $= \frac{- 2}{3}$ [compare it with $y = m x + c$ ]

Hence, $m = \frac{- 2}{3} . . . . . . . . . . . . (1)$

Given curve is $x^{2} + y^{2} = 13$

Differentiate both side with respect to $x$

$2 x + 2 y \cdot \frac{d y}{d x} = 0$

At point $(x_{1}, y_{1})$

$\Rightarrow \frac{d y}{d x} = \frac{- x_{1}}{y_{1}}$

Let slope of tangent at $(x_{1}, y_{1})$ be $m$

Then $m = \frac{- x_{1}}{y_{1}} . . . . . . . . . . . . . (2)$

Compare equation $(1)$ and $(2)$

$\frac{- x_{1}}{y_{1}} = \frac{- 2}{3}$

$\Rightarrow x_{1} = \frac{2}{3} y_{1} . . . . . . . . . . . . . . (3)$

Also point $(x_{1}, y_{1})$ lies on the curve, So

${x_{1}}^{2} + {y_{1}}^{2} = 13 . . . . . . . . . . . (4)$

Now, put value of $x_{1}$ from $(3)$ and $(4)$

${(\frac{2}{3} y_{1})}^{2} + {y_{1}}^{2} = 13$

$\Rightarrow \frac{4 {y_{1}}^{2}}{9} + {y_{1}}^{2} = 13$

$\Rightarrow 13 {y_{1}}^{2} = 13 \times 9$

$\Rightarrow {y_{1}}^{2} = 9$

$\Rightarrow y_{1} = \pm 3$

From equation $(3)$ $x_{1} = \frac{2}{3} y_{1}$

So when, $y_{1} = 3, x_{1} = 2$

And when $y_{1} = - 3, x_{1} = - 2$

Hence, points are $(2, 3)$ and $(- 2, - 3)$ .

Find the points on the curve $x^{2} + y^{2} = 13$ , the tangent at each one of which is parallel to $2 x + 3 y = 7$ .

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the points on the curve x2+y2=13, the tangent at each one of which is parallel to 2x+3y=7.

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the points on the curve $x^{2} + y^{2} = 13$ , the tangent at each one of which is parallel to $2 x + 3 y = 7$ .