The differential equation for all the straight lines which are at a unit distance from the origin is

Question

The differential equation for all the straight lines which are at a unit distance from the origin is (A) (y-x (d y/d x))2=1-((d y/d x))2 (B) (y+x (d y/d x))2=1+((d y/d x))2 (C) (y-x (d y/d x))2=1+((d y/d x))2 (D) (y+x (d y/d x))2=1-((d y/d x))2

Tardigrade Admin · Accepted Answer

Let equation of St. Line Y-y=m(X-x) Distance from origin ⇒|( mx - y /√1+ m 2)|=1 ∴ (m x-y)2=1+m2 (y-(d y/d x) x)2=1+((d y/d x))2

Q. The differential equation for all the straight lines which are at a unit distance from the origin is

$(y - x \frac{d y}{d x})^{2} = 1 - (\frac{d y}{d x})^{2}$

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

$(y - x \frac{d y}{d x})^{2} = 1 + (\frac{d y}{d x})^{2}$

$(y + x \frac{d y}{d x})^{2} = 1 - (\frac{d y}{d x})^{2}$

Let equation of St. Line
$Y - y = m (X - x)$
Distance from origin $\Rightarrow ∣ ∣ \frac{m x - y}{1 + m ^{2}} ∣ ∣ = 1$
$∴ (m x - y)^{2} = 1 + m^{2}$
$(y - \frac{d y}{d x} x)^{2} = 1 + (\frac{d y}{d x})^{2}$

Q. The differential equation for all the straight lines which are at a unit distance from the origin is

(y−xdxdy​)2=1−(dxdy​)2

(y+xdxdy​)2=1+(dxdy​)2

(y−xdxdy​)2=1+(dxdy​)2

(y+xdxdy​)2=1−(dxdy​)2

Let equation of St. Line Y−y=m(X−x) Distance from origin ⇒∣∣​1+m2​mx−y​∣∣​=1 ∴(mx−y)2=1+m2 (y−dxdy​x)2=1+(dxdy​)2

$(y - x \frac{d y}{d x})^{2} = 1 - (\frac{d y}{d x})^{2}$

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

$(y - x \frac{d y}{d x})^{2} = 1 + (\frac{d y}{d x})^{2}$

$(y + x \frac{d y}{d x})^{2} = 1 - (\frac{d y}{d x})^{2}$

Let equation of St. Line
$Y - y = m (X - x)$
Distance from origin $\Rightarrow ∣ ∣ \frac{m x - y}{1 + m ^{2}} ∣ ∣ = 1$
$∴ (m x - y)^{2} = 1 + m^{2}$
$(y - \frac{d y}{d x} x)^{2} = 1 + (\frac{d y}{d x})^{2}$