The equation of the circle passing through the foci of the ellipse $\dfrac{{{x^2}}}{{16}} + \dfrac{{{y^2}}}{9} = 1$ and having centre at (0,3) is
A.${x^2} + {y^2} - 6y - 7 = 0$
B.${x^2} + {y^2} - 6y + 7 = 0$
C.${x^2} + {y^2} - 6y - 5 = 0$
D.${x^2} + {y^2} - 6y + 5 = 0$
Answer
327.3k+ views
Hint-In this question we have to find the equation of the circle passing through the foci of the ellipse $\dfrac{{{x^2}}}{{16}} + \dfrac{{{y^2}}}{9} = 1$ but one must know briefly that ellipse is a plane curve surrounding two focal points and ellipse is determined by its foci as such, it generalises a circle but if you can use the lengths of the major and the minor axes to find its coordinates.
Complete step-by-step answer:
Before we go for the solution of this sum we must know the meaning of ellipse and foci of ellipse so in mathematics, ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal point is a constant. As such, it generalizes a circle, which is the special type of ellipse in which two focal points are the same and we should know that an ellipse is determined by its foci. But if you determine you can use the lengths of the major and the minor axes to find its coordinates. Now, the solution of the question is:
Coordinates of Foci are $( \pm ae,0)$
$a = 4,b = 3,e = \sqrt {1 - \dfrac{9}{{16}}} = \dfrac{{\sqrt 7 }}{4}$
Radius of the circle = $\sqrt {{{(ae)}^2} + {3^2}} $ =$\sqrt {16} = 4$
Equation of circle:
$
{\left( {x - 0} \right)^2} + {\left( {y - 3} \right)^2} = 16 \\
\Rightarrow {x^2} + {y^2} - 6y - 7 = 0 \\
$
Hence, option A is correct.
Note-In this question it is to be noted that for finding the equation of the circle passing through the foci of ellipse one should know the meaning of ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal point is a constant. As such it generalizes a circle. an ellipse is determined by its foci but if you determine you can use the lengths of the major and minor axes to find its coordinates. It should be noticed that I have used this definition a lot of times this is because one must clear his basics before moving on. Now, for a solution I have simply used the given values in the question to calculate the equation and simplify it.
Complete step-by-step answer:
Before we go for the solution of this sum we must know the meaning of ellipse and foci of ellipse so in mathematics, ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal point is a constant. As such, it generalizes a circle, which is the special type of ellipse in which two focal points are the same and we should know that an ellipse is determined by its foci. But if you determine you can use the lengths of the major and the minor axes to find its coordinates. Now, the solution of the question is:
Coordinates of Foci are $( \pm ae,0)$
$a = 4,b = 3,e = \sqrt {1 - \dfrac{9}{{16}}} = \dfrac{{\sqrt 7 }}{4}$
Radius of the circle = $\sqrt {{{(ae)}^2} + {3^2}} $ =$\sqrt {16} = 4$
Equation of circle:
$
{\left( {x - 0} \right)^2} + {\left( {y - 3} \right)^2} = 16 \\
\Rightarrow {x^2} + {y^2} - 6y - 7 = 0 \\
$
Hence, option A is correct.
Note-In this question it is to be noted that for finding the equation of the circle passing through the foci of ellipse one should know the meaning of ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal point is a constant. As such it generalizes a circle. an ellipse is determined by its foci but if you determine you can use the lengths of the major and minor axes to find its coordinates. It should be noticed that I have used this definition a lot of times this is because one must clear his basics before moving on. Now, for a solution I have simply used the given values in the question to calculate the equation and simplify it.
Last updated date: 02nd Jun 2023
•
Total views: 327.3k
•
Views today: 5.84k
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

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

Tropic of Cancer passes through how many states? Name them.

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

Name the Largest and the Smallest Cell in the Human Body ?
