Answer

Verified

338.7k+ 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

Basicity of sulphurous acid and sulphuric acid are

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

What is the stopping potential when the metal with class 12 physics JEE_Main

The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main

Using the following information to help you answer class 12 chemistry CBSE

Why should electric field lines never cross each other class 12 physics CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

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

Describe the poetic devices used in the poem Aunt Jennifers class 12 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

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

State the laws of reflection of light

State and prove Bernoullis theorem class 11 physics CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE