Show that \[{x^2} - 6x + 5{y^2} + 10x - 14y + 9 = 0\]. Find the acute angle?
Answer
278.4k+ views
Hint: The general equation of a straight line is \[y = mx + c\], where m is the gradient, and y = c is the value where the line cuts the y-axis. This number c is called the intercept on the y-axis. The equation of a straight line with gradient m and intercept c on the y-axis is \[y = mx + c\].
Complete step-by-step solution:
Given
\[\therefore {x^2} - 6x + 5{y^2} + 10x - 14y + 9 = 0\]
Comparing with
$\therefore a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$
We get that
$a = 1,\,h = - 3,\,\,b = 5,\,g = 5,\,f = - 7,\,c = 9$
We know that
$\therefore \left[ {\begin{array}{*{20}{c}}
a&h&g \\
h&b&f \\
g&f&c
\end{array}} \right]$
.Put the value
$ = \left[ {\begin{array}{*{20}{c}}
1&{ - 3}&5 \\
{ - 3}&5&{ - 7} \\
5&{ - 7}&9
\end{array}} \right]$
Simplify
$ = 1\{ (9 \times 5) - ( - 7 \times - 7)\} + 3\{ (9 \times - 3) - (5 \times - 7)\} + 5\{ ( - 3 \times - 7) - (5 \times 5)\} $
$ = (45 - 49) + 3( - 27 + 35) + 5(21 - 25)$
$ = - 4 + 3 \times 8 + 5 \times - 4$
$ = 24 - 24$
$ = 0$
Given equation present the equation of line
Now
As we know that
$\therefore \tan \theta = \left[ {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right]$
Put the value
$ \Rightarrow \tan \theta = \left[ {\dfrac{{2\sqrt {{{( - 3)}^2} - (1 \times 5)} }}{{1 + 5}}} \right]$
$ \Rightarrow \tan \theta = \left[ {\dfrac{{2\sqrt {9 - 5} }}{6}} \right]$
$ \Rightarrow \tan \theta = \left[ {\dfrac{{\sqrt 4 }}{3}} \right]$
$ \Rightarrow \tan \theta = \left( {\dfrac{2}{3}} \right)$
$ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{2}{3}} \right)$
Note: The slope of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter ‘m’. The line is increasing. It goes up from left to right. The slope is positive. A line is decreasing if it goes down from left to right. The slope is negative. If a line is horizontal the slope is zero. This is a constant function.
Complete step-by-step solution:
Given
\[\therefore {x^2} - 6x + 5{y^2} + 10x - 14y + 9 = 0\]
Comparing with
$\therefore a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$
We get that
$a = 1,\,h = - 3,\,\,b = 5,\,g = 5,\,f = - 7,\,c = 9$
We know that
$\therefore \left[ {\begin{array}{*{20}{c}}
a&h&g \\
h&b&f \\
g&f&c
\end{array}} \right]$
.Put the value
$ = \left[ {\begin{array}{*{20}{c}}
1&{ - 3}&5 \\
{ - 3}&5&{ - 7} \\
5&{ - 7}&9
\end{array}} \right]$
Simplify
$ = 1\{ (9 \times 5) - ( - 7 \times - 7)\} + 3\{ (9 \times - 3) - (5 \times - 7)\} + 5\{ ( - 3 \times - 7) - (5 \times 5)\} $
$ = (45 - 49) + 3( - 27 + 35) + 5(21 - 25)$
$ = - 4 + 3 \times 8 + 5 \times - 4$
$ = 24 - 24$
$ = 0$
Given equation present the equation of line
Now
As we know that
$\therefore \tan \theta = \left[ {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right]$
Put the value
$ \Rightarrow \tan \theta = \left[ {\dfrac{{2\sqrt {{{( - 3)}^2} - (1 \times 5)} }}{{1 + 5}}} \right]$
$ \Rightarrow \tan \theta = \left[ {\dfrac{{2\sqrt {9 - 5} }}{6}} \right]$
$ \Rightarrow \tan \theta = \left[ {\dfrac{{\sqrt 4 }}{3}} \right]$
$ \Rightarrow \tan \theta = \left( {\dfrac{2}{3}} \right)$
$ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{2}{3}} \right)$
Note: The slope of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter ‘m’. The line is increasing. It goes up from left to right. The slope is positive. A line is decreasing if it goes down from left to right. The slope is negative. If a line is horizontal the slope is zero. This is a constant function.
Recently Updated Pages
Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How many meters are there in a kilometer And how many class 8 maths CBSE

What is pollution? How many types of pollution? Define it

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

What were the major teachings of Baba Guru Nanak class 7 social science CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a labelled sketch of the human eye class 12 physics CBSE
