Question

If the curve $ay+x^2=7$, and $x^3=y$ cut orthogonally at $\left(1,1\right)$, then the value of $a$ is

• A. $1$
• B. $0$
• C. $6$
• D. 6

Answer 1

The correct option is C

$6$

Step 1: Find the slope $\frac{dy}{dx}$ at any point on given curves

Equation of the curve are

$ay+x^2=7$ $...........................$$\left(1\right)$

$x^3=y$ $...........................$$\left(2\right)$

Differentiating the equation $\left(1\right)$ with respect to $x$

$$\begin{gathered} a\frac{dy}{dx}+2x=0 \\ \Rightarrow \frac{dy}{dx}=-\frac{2x}{a} \\ \therefore m_1=-\frac{2x}{a} \end{gathered}$$

Differentiating the equation $\left(2\right)$ with respect to $x$

$$\begin{gathered} 3x^2=\frac{dy}{dx} \\ \because m_2=3x^2 \end{gathered}$$

Step 2: Apply condition of orthogonality and solve equation to find a

Curves cuts orthogonally

$$\begin{gathered} m_1\times m_2=-1 \\ \Rightarrow -\frac{2x}{a}\times 3x^2=-1 \\ \Rightarrow \frac{6x^3}{a}=1 \\ \Rightarrow 6x^3=a \end{gathered}$$

Since $\left(1,1\right)$ is the point of intersection of two curves

Put $x=1$

$$\begin{gathered} a=6·{\left(1\right)}^3 \\ \Rightarrow a=6 \end{gathered}$$

Hence, $a=6$, so, the correct option is (C).