Animated Solution for Mathematics - Differentiation: If the curves $y^2=6x,9x^2+b{y}^2=16$ intersect each other at right angles, then the value of $b$ is :

Imagine you are standing on a coordinate plane, watching two paths cross. One is a parabola, $y^2=6x$, sweeping gracefully to the right. The other is an ellipse, $9x^2+b{y}^2=16$, a closed loop waiting to be defined by the parameter $b$.

When we say these curves intersect 'orthogonally,' we are describing a moment of perfect perpendicularity. At the exact point $(x_1,y_1)$ where they cross, the tangent lines to the parabola and the ellipse form a perfect $90^{\circ }$ angle.

In the language of calculus, if $m_1$ is the slope of the first curve and $m_2$ is the slope of the second, the condition for this perpendicular intersection is simply $m_1\cdot m_2=-1$.

Finding the Slopes

To unlock the value of $b$, we first determine the slopes of both curves. Starting with the parabola $y^2=6x$, we differentiate both sides with respect to $x$.

Using the chain rule, the derivative of $y^2$ is $2y\frac{dy}{dx}$, and the derivative of $6x$ is $6$. Thus, we have:

Solving for the slope $m_1$ at the point $(x_1,y_1)$, we obtain:

Now, we turn our attention to the ellipse $9x^2+b{y}^2=16$. Applying implicit differentiation, the derivative of $9x^2$ is $18x$, and the derivative of $b{y}^2$ is $2by\frac{dy}{dx}$.

Setting the derivative of the constant $16$ to $0$, we get:

Rearranging this to isolate the slope $m_2$, we find:

Eliminating the Unknowns

We now combine our two slopes into the orthogonality condition $m_1\cdot m_2=-1$. Substituting our expressions, we have:

Multiplying these fractions yields:

Since the point $(x_1,y_1)$ lies on the parabola, it must satisfy the equation $y_1^2=6x_1$. Substituting this into our expression allows us to eliminate the variables:

Assuming $x_1 eq 0$, we cancel $x_1$ from the numerator and denominator to get:

A quick cross-multiplication gives $6b=27$. Dividing by $6$ and simplifying the fraction, we arrive at our final answer: