Trace the following central conics.
\[3{(2x - 3y + 4)^2} + 2{(3x + 2y - 5)^2} = 78.\]
Last updated date: 16th Mar 2023
•
Total views: 307.5k
•
Views today: 2.86k
Answer
307.5k+ views
Hint: Since this equation is similar to that of ellipse hence we will compare this equation with
the standard equation of ellipse.
The standard equation of ellipse is:
$\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$
Now comparing this equation with our given equation we see that we have an ellipse
\[3{(2x - 3y + 4)^2} + 2{(3x + 2y - 5)^2} = 78.\]
Now multiplying and dividing LHS by $13$ we have
$\dfrac{{3 \times 13{{(2x - 3y + 4)}^2}}}{{{{(\sqrt {13} )}^2}}} + \dfrac{{2 \times 13{{(3x + 2y - 5)}^2}}}{{{{(\sqrt {13} )}^2}}} = 78$
Now let $X = \dfrac{{2x - 3y + 4}}{{\sqrt {13} }},Y = \dfrac{{3x + 2y - 5}}{{\sqrt {13} }}$
on substituting it in our equation we have
$39{X^2} + 26{Y^2} = 78$
Now dividing both sides by $78$ we get
$\dfrac{{{X^2}}}{2} + \dfrac{{{Y^2}}}{3} = 1$
on comparing this equation with the standard equation of ellipse we get
$a = \sqrt 2 ,b = \sqrt 3 $
Note: While attempting question on conic sections and especially locus questions we should always compare the given equation in the question with standard equations we know of various conics because the equation in the question is always the modified equation of any of the conics and hence by comparing it and knowing of which conic it is we can further continue by then converting it to the standard form and we finally arrive to solution by this method
the standard equation of ellipse.
The standard equation of ellipse is:
$\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$
Now comparing this equation with our given equation we see that we have an ellipse
\[3{(2x - 3y + 4)^2} + 2{(3x + 2y - 5)^2} = 78.\]
Now multiplying and dividing LHS by $13$ we have
$\dfrac{{3 \times 13{{(2x - 3y + 4)}^2}}}{{{{(\sqrt {13} )}^2}}} + \dfrac{{2 \times 13{{(3x + 2y - 5)}^2}}}{{{{(\sqrt {13} )}^2}}} = 78$
Now let $X = \dfrac{{2x - 3y + 4}}{{\sqrt {13} }},Y = \dfrac{{3x + 2y - 5}}{{\sqrt {13} }}$
on substituting it in our equation we have
$39{X^2} + 26{Y^2} = 78$
Now dividing both sides by $78$ we get
$\dfrac{{{X^2}}}{2} + \dfrac{{{Y^2}}}{3} = 1$
on comparing this equation with the standard equation of ellipse we get
$a = \sqrt 2 ,b = \sqrt 3 $
Note: While attempting question on conic sections and especially locus questions we should always compare the given equation in the question with standard equations we know of various conics because the equation in the question is always the modified equation of any of the conics and hence by comparing it and knowing of which conic it is we can further continue by then converting it to the standard form and we finally arrive to solution by this method
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
