Statement -1: The slope of the tangent at any | vedclass

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(B) Statement $-1$: For a parabola with vertex at the origin and axis along the $x$-axis,the equation is $y^2 = 4ax$ (or $y^2 = -4ax$).Differentiating

Vedclass · Accepted Answer

Statement $-1$ is true; Statement $-2$ is true; Statement $-2$ is not a correct explanation for Statement $-1$.

Vedclass · Answer

(B) Statement $-1$: For a parabola with vertex at the origin and axis along the $x$-axis,the equation is $y^2 = 4ax$ (or $y^2 = -4ax$).Differentiating with respect to $x$,we get $2y \frac{dy}{dx} = 4a$,which implies $\frac{dy}{dx} = \frac{2a}{y}$.Thus,the slope $\frac{dy}{dx}$ is inversely proportional to the ordinate $y$. So,Statement $-1$ is true.Statement $-2$: The differential equation for the family $y^2 = 4ax$ is obtained by eliminating $a$: $a = \frac{y^2}{4x}$. Substituting this into $2y \frac{dy}{dx} = 4a$ gives $2y \frac{dy}{dx} = 4(\frac{y^2}{4x}) = \frac{y^2}{x}$,or $2x \frac{dy}{dx} = y$.This is a first-order,first-degree differential equation. So,Statement $-2$ is true.However,Statement $-2$ describes the differential equation of the family,while Statement $-1$ is a property of the tangent slope derived from the equation of the parabola. Statement $-2$ is not the reason for the property in Statement $-1$.Therefore,both are true,but Statement $-2$ is not the correct explanation for Statement $-1$.

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