Question

How do you write an equation for the translation of $x^2+y^2$ = $25$ by 7 units left and 2 units down?

Answer 1

Hint: This question is from the topic of geometry. In solving this question, we will first understand the given equation using the figure. After that, we will shift the equation by 7 units left. After that, we will shift the equation by 2 units down. After solving the further question, we will get our answer.

Complete step by step answer:
Let us solve this question.
First, we will write the given equation from the question $x^2+y^2$ = $25$ in proper format. So, we have the equation given as \[{{x}^{2}}+{{y}^{2}}=25\]. We have asked to find the equation, if we shift the equation 7 units left and 2 units down using the given equation.
The graph of \[{{x}^{2}}+{{y}^{2}}=25\] will be:

Now, it is saying that the graph is shifting left by 7 units. Then, we will add 7 to x in the equation \[{{x}^{2}}+{{y}^{2}}=25\].
Then, the new equation will be \[{{\left( x+7 \right)}^{2}}+{{y}^{2}}=25\] and the new graph will be:

According to the question, we shifted the graph to the left side by 7 units. Now, we will shift the graph down by 2 units.
For shifting down by 2 units, we will add 2 to y in the equation \[{{\left( x+7 \right)}^{2}}+{{y}^{2}}=25\].
So, the new equation will be \[{{\left( x+7 \right)}^{2}}+{{\left( y+2 \right)}^{2}}=25\] and the new graph will be:

So, we can say that the equation for the translation of \[{{x}^{2}}+{{y}^{2}}=25\] by 7 units lefts and 2 units down will be \[{{\left( x+7 \right)}^{2}}+{{\left( y+2 \right)}^{2}}=25\].