Name: Use of Derivative in Rectilinear Motion
Uploaded: 2019-07-19
Duration: 3 min 43 s
Description: Do you know we can use the derivatives to know the velocity and acceleration of a particle? Watch this video discussing the uses of derivatives in Rectilinea...

Question

Let a curve

y = f (x)

pass through

(1, 1)

. At any point

P

on the curve, tangent and normal to the curve are drawn to intersect the

x

-axis at

Q

and

R

respectively. If

Q R = 2

. Find the equation of all such possible curves.

Accepted Answer

Curve $y = f (x)$ passes through the point $(1, 1)$

Let at any point $P (x, y)$ on curve, tangent and normal are drawn that intersect $x$ -axis at $Q$ and $R$ ,

$\Rightarrow Q R = 2$

Slope of tangent $= \frac{d y}{d x}$

Slope of normal is $= \frac{- 1}{\frac{d y}{d x}}$

Equation of tangent: $Y - y = \frac{d y}{d x} (X - x)$ ..... $(1)$

Equation of normal: $Y - y = \frac{- d x}{d y} (X - x)$ ...... $(2)$

Tangent intersects $x$ -axis at $Q$ , put $Y = 0$ in equation $(1)$ ,

$X = x - y \frac{d x}{d y}$

$Q (x - y \frac{d x}{d y}, 0)$

Normal intersects $x$ -axis at $R$ , put $Y = 0$ in equation $(2)$ ,

$X = x + y \frac{d y}{d x}, R (x + y \frac{d y}{d x}, 0)$

$Q R = 2$

$(x + y \frac{d y}{d x}) - (x - y \frac{d x}{d y}) = 2$

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

$y \frac{d y}{d x} + y \frac{1}{\frac{d y}{d x}} = 2$

$y {(\frac{d y}{d x})}^{2} - 2 \frac{d y}{d x} + y = 0$

On solving,

$\frac{d y}{d x} = \frac{1 \pm \sqrt{1 - y^{2}}}{y}$

$\frac{y d y}{1 \pm \sqrt{1 - y^{2}}} = d x$

On multiplying by conjugate of $1 \pm \sqrt{1 - y^{2}}$ in denominator and numerator,

$\frac{1 \mp \sqrt{1 - y^{2}}}{y} d y = d x$

$\frac{1}{y} d y \mp \frac{\sqrt{1 - y^{2}}}{y} d y = d x$

On Integrating both sides,

$\log_{e} y \mp (\log_{e} |\frac{1 - \sqrt{1 - y^{2}}}{y}| + \sqrt{1 - y^{2}}) = x + C$

It passes through $(1, 1)$

On substituting values of $x$ and $y$ ,

$C = - 1 \log_{e} y = \mp (\log_{e} |\frac{1 - \sqrt{1 - y^{2}}}{y}| + \sqrt{1 - y^{2}}) + x - 1$

Let a curve $y = f (x)$ pass through $(1, 1)$ . At any point $P$ on the curve, tangent and normal to the curve are drawn to intersect the $x$ -axis at $Q$ and $R$ respectively. If $Q R = 2$ . Find the equation of all such possible curves.

Important Questions on dy/dx as a Rate Measurer & Tangents, Normals

Learn and Prepare for any exam you want !

Let a curve y=fx pass through 1,1. At any point P on the curve, tangent and normal to the curve are drawn to intersect the x-axis at Q and R respectively. If QR=2. Find the equation of all such possible curves.

Important Questions on dy/dx as a Rate Measurer & Tangents, Normals

Learn and Prepare for any exam you want !

Let a curve $y = f (x)$ pass through $(1, 1)$ . At any point $P$ on the curve, tangent and normal to the curve are drawn to intersect the $x$ -axis at $Q$ and $R$ respectively. If $Q R = 2$ . Find the equation of all such possible curves.