Question

If $y^2=a{x}^2+bx+c$, then $y^3.\frac{d^{2}y}{d{x}^2}$ is

• A. A constant
• B. A function of x only
• C. A function of y only
• D. A function of x and y

Answer 1

The correct option is A A constant

$y^2=a{x}^2+bx+c$ .......... (1)

Differentiate w.r.t $x$

$2y{y}_1=2ax+b$ .......... (2)
Here, $y_1=\frac{dy}{dx}$

Again, diff w.r.t to $x$

$2y_1^2+2y{y}_2=2a$

$y_1^2+y{y}_2=a$ ............. (3)

$\Rightarrow y^2{y}_1^2+y^3{y}_2=a(a{x}^2+bx+c)$ using (2)

$\Rightarrow {(ax+\frac{b}{2})}^2+y^3{y}_2=a^2{x}^2+abx+ac$

$\Rightarrow a^2{x}^2+abx+\frac{b^2}{4}+y^3{y}_2=a^2{x}^2+abx+ac$

$\Rightarrow y^3{y}_2=ac-\frac{b^2}{4}$

$y^3{y}_2=\frac{4ac-b^2}{4}$ is a constant.