Answer

Verified

387.6k+ views

**Hint:**This question belongs to the topic of pre-calculus. In this question, first we will group the terms of x in one bracket and group the terms of y in another bracket. After that, we will add the terms to make the perfect square which are in the brackets and also we will subtract the terms outside the bracket to balance the equation. After that, we will solve the further equation and get the answer in standard form.

**Complete step by step solution:**

Let us solve this question.

In this question, we have asked to convert the given equation in the standard form. The given equation is \[4{{x}^{2}}-{{y}^{2}}-24x+4y+28=0\]. This equation can also be written as

\[\Rightarrow \left( 4{{x}^{2}}-24x \right)-\left( {{y}^{2}}-4y \right)+28=0\]

Now, we will add a third term in both the brackets to make a complete square of that bracket. We will add the term to make a perfect square in the form of \[{{a}^{2}}-2\times a\times b+{{b}^{2}}\] in the first bracket and also we will add the term to make the perfect square in the form of \[{{a}^{2}}-2\times a\times b+{{b}^{2}}\] in the second bracket. And, also we will subtract the terms to balance the equation.

So, we can write the above equation as

\[\Rightarrow \left( 4{{x}^{2}}-24x+{{\left( \dfrac{24}{2\times 2} \right)}^{2}} \right)-\left( {{y}^{2}}-4y+{{\left( \dfrac{4}{2} \right)}^{2}} \right)+28-{{\left( \dfrac{24}{2\times 2} \right)}^{2}}-\left( -{{\left( \dfrac{4}{2} \right)}^{2}} \right)=0\]

We can write the above equation as

\[\Rightarrow \left( 4{{x}^{2}}-24x+36 \right)-\left( {{y}^{2}}-4y+4 \right)+28-36+4=0\]

The above equation can also be written as

\[\Rightarrow \left( {{\left( 2x \right)}^{2}}-2\times 2x\times 6+{{\left( 6 \right)}^{2}} \right)-\left( {{y}^{2}}-2\times y\times 2+{{\left( 2 \right)}^{2}} \right)+28-36+4=0\]

The above equation can also be written as

\[\Rightarrow \left( {{\left( 2x \right)}^{2}}-2\times 2x\times 12+{{\left( 6 \right)}^{2}} \right)-\left( {{y}^{2}}-2\times y\times 2+{{\left( 2 \right)}^{2}} \right)-4=0\]

Using the formula \[{{a}^{2}}-2\times a\times b+{{b}^{2}}={{\left( a-b \right)}^{2}}\], we can write the above equation as

\[\Rightarrow {{\left( 2x-6 \right)}^{2}}-{{\left( y-2 \right)}^{2}}-4=0\]

Now, taking common out 2 from the first bracket of the equation, we get

\[\Rightarrow {{2}^{2}}\times {{\left( x-3 \right)}^{2}}-{{\left( y-2 \right)}^{2}}-4=0\]

We can write the above equation as

\[\Rightarrow 4{{\left( x-3 \right)}^{2}}-{{\left( y-2 \right)}^{2}}=4\]

Now, dividing 4 to the both side of the equation, we get

\[\Rightarrow {{\left( x-3 \right)}^{2}}-\dfrac{{{\left( y-2 \right)}^{2}}}{4}=1\]

Hence, we get that the standard form of the equation \[4{{x}^{2}}-{{y}^{2}}-24x+4y+28=0\] is \[{{\left( x-3 \right)}^{2}}-\dfrac{{{\left( y-2 \right)}^{2}}}{4}=1\]

**Note:**

As we can see that this question is from the topic of pre-calculus, so we should have a better knowledge in that topic for solving this type of question. We should remember the formula that is \[{{a}^{2}}-2\times a\times b+{{b}^{2}}={{\left( a-b \right)}^{2}}\] to solve this type of question easily. We can see that this equation is in the form of hyperbola. The general equation of hyperbola is always in the form of \[\dfrac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}-\dfrac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1\].

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Change the following sentences into negative and interrogative class 10 english CBSE