The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is

Question

The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is (A) an ellipse (B) parabola (C) circle (D) rectangular hyperbola

Tardigrade Admin · Accepted Answer

Slope of tangent, (dy/dx)=(x/y) ⇒ ydx=xdx On integrating, we get (y2/2)=(x2/2)+k ⇒ y2-x2=2k=c (say) which is equation of a rectangular hyperbola.

Q. The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is

an ellipse

parabola

circle

rectangular hyperbola

Slope of tangent, $\frac{d y}{d x} = \frac{x}{y}$
$\Rightarrow y d x = x d x$
On integrating, we get
$\frac{y ^{2}}{2} = \frac{x ^{2}}{2} + k$
$\Rightarrow y^{2} - x^{2} = 2 k = c$ (say)
which is equation of a rectangular hyperbola.

Q. The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is

an ellipse

parabola

circle

rectangular hyperbola

Slope of tangent, dxdy​=yx​ ⇒ydx=xdx On integrating, we get 2y2​=2x2​+k ⇒y2−x2=2k=c (say) which is equation of a rectangular hyperbola.

Slope of tangent, $\frac{d y}{d x} = \frac{x}{y}$
$\Rightarrow y d x = x d x$
On integrating, we get
$\frac{y ^{2}}{2} = \frac{x ^{2}}{2} + k$
$\Rightarrow y^{2} - x^{2} = 2 k = c$ (say)
which is equation of a rectangular hyperbola.