Question

At any point on the curve $2x^2{y}^2-x^4=c$, the mean proportional between the abscissa and the difference between the abscissa and the sub-normal drawn to the curve at the same point is equal to

• A. ordinate
• B. radius vector
• C. x-intercept of tangent
• D. sub-tangent

Answer 1

Answer: (A)

ordinate

$2x^2{y}^2-x^4=c$

$\frac{dy}{dx}=\frac{x^2-y^2}{xy}$

Slope of tangent at $P(x_1,y_1)ism=\frac{x_1^2-y_1^2}{x_1{y}_1}$

Length of sub-normal $=|y_{1}m|=\frac{x_1^2-y_1^2}{x_1}$

Now, difference between abscissa and length of sub-normal $=x_1-\frac{x_1^2-y_1^2}{x_1}=\frac{y_1^2}{x_1}$

Required Mean proportional $=\sqrt{x_1\frac{y_1^2}{x_1}}=y_1$