Question

How do you find the intercepts of ${{x}^{2}}y-{{x}^{2}}+4y=0$ ?

Answer 1

Hint: Whenever we talk about intercepts, we are talking about where the x and y axes are intersecting. Thus, we shall first modify the equation further by taking the terms common. Then we will put the value of x-variable and y-variable equal to zero in the equation one by one to find the required intercepts.

Complete step-by-step solution:
The x-intercept is the distance from origin of the point on the given function where the value of y is zero. This point logically lies on the x-axis and is given as $\left( a,0 \right)$ where $a$ is called the x-intercept.
The y-intercept is the distance from origin of the point on the given function where the value of x is zero. This point logically lies on the y-axis and is given as $\left( 0,b \right)$ where $b$ is called the y-intercept.
We are given the function, ${{x}^{2}}y-{{x}^{2}}+4y=0$.
Taking y common in the given expression and separating the terms of the x-variable and y-variable, we get
$\begin{align}
  & \Rightarrow y\left( {{x}^{2}}+4 \right)={{x}^{2}} \\
 & \Rightarrow y=\dfrac{{{x}^{2}}}{{{x}^{2}}+4} \\
\end{align}$
In order to find the x-intercept, we will put $y=0$ and solve the equation accordingly. Hence, putting $y=0$, we get
$\begin{align}
  & \Rightarrow 0=\dfrac{{{x}^{2}}}{{{x}^{2}}+4} \\
 & \Rightarrow {{x}^{2}}=0 \\
\end{align}$
Taking square root, we get
$\Rightarrow x=0$
Therefore, the x-intercept is equal to 0.
In order to find the y-intercept, we will put $x=0$ and solve the equation accordingly. Hence, putting $x=0$, we get
$\begin{align}
  & \Rightarrow y=\dfrac{{{0}^{2}}}{{{0}^{2}}+4} \\
 & \Rightarrow y=0 \\
\end{align}$
Therefore, the y-intercept is equal to 0.
Hence, the x-intercept as well as the y-intercept is equal to 0.

Note: Sometimes we are not able to manipulate the given complex mathematical equation according to the question and get the required solutions. Thus, we must have an ample practice of problems in order to recognize the terms that can be taken common or the changes that be made to the equation to make it simpler.