Problem 88 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Equation of the ellipse which passes through the point \((-3,1)\), whose axes are the coordinate axes and has eccentricity \(\sqrt{\frac{2}{5}}\) is (A) \(5 x^{2}+3 y^{2}-48=0\) (B) \(3 x^{2}+5 y^{2}-15=0\) (C) \(5 x^{2}+3 y^{2}-32=0\) (D) \(3 x^{2}+5 y^{2}-32=0\)

Step-by-Step Solution

Answer

The correct equation is option (A): \(5 x^{2}+3 y^{2}-48=0\).

Step 1: Understand the standard form of an ellipse equation

The equation of an ellipse with semi-major axis along the x-axis and semi-minor axis along the y-axis, centered at the origin, is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

Step 2: Relate eccentricity to semi-major and semi-minor axes

The eccentricity \(e\) of an ellipse is given by \( e = \sqrt{1 - \frac{b^2}{a^2}} \). Substituting the given eccentricity \(e = \sqrt{\frac{2}{5}}\), we get \( \frac{2}{5} = 1 - \frac{b^2}{a^2} \). Solving this gives \( \frac{b^2}{a^2} = \frac{3}{5} \).

Step 3: Use the point on the ellipse

The ellipse passes through the point \((-3, 1)\), which implies \( \frac{(-3)^2}{a^2} + \frac{(1)^2}{b^2} = 1 \). Substituting \( b^2 = \frac{3}{5}a^2 \) from Step 2, \( \frac{9}{a^2} + \frac{1}{(\frac{3}{5}a^2)} = 1 \).

Step 4: Simplify the equation using known relations

Multiply both sides of the equation from Step 3 by \( a^2 \): \( 9 + \frac{5}{3} = a^2 \). Solving for \( a^2 \) gives \( a^2 = \frac{48}{5} \).

Step 5: Determine the values of \(a^2\) and \(b^2\)

Since \( \frac{b^2}{a^2} = \frac{3}{5} \), \( b^2 = \frac{3}{5} \times \frac{48}{5} = \frac{144}{25} \).

Step 6: Write the ellipse equation

Substituting \( a^2 \) and \( b^2 \) into the ellipse equation: \( \frac{x^2}{\frac{48}{5}} + \frac{y^2}{\frac{144}{25}} = 1 \). Multiply through by \(5 \times \frac{25}{144}\) to get the integer coefficient equation: \( 5x^2 + 3y^2 = 48 \).

Step 7: Solve for the given options

Rewriting the equation obtained: \( 5 x^2 + 3 y^2 = 48 \) matches with option (A) after transforming to \( 5 x^{2}+3 y^{2}-48=0 \).

Key Concepts

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being circular. In simple terms, it's a number that tells you how 'stretched out' an ellipse is.
For any ellipse, the eccentricity value ranges from 0 to 1.

• An eccentricity of 0 means the ellipse is actually a perfect circle.
• A value closer to 1 indicates a very elongated shape.

To find the eccentricity of an ellipse given the semi-major axis (\( a \)) and semi-minor axis (\( b \)), we use the formula: \[ e = \sqrt{1 - \frac{b^2}{a^2}} \] In the given problem, the eccentricity is provided as \( e = \sqrt{\frac{2}{5}} \)This tells us that the ellipse is neither a perfect circle nor too elongated.
By substituting into the eccentricity formula, we relate the semi-major and semi-minor axes, which is crucial in determining the complete equation of the ellipse.

Semi-major axis

The semi-major axis is one of the most defining dimensions of an ellipse. It is one-half of the longest diameter of the ellipse and is conventionally denoted by \( a \).
In the equation of an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the semi-major axis is crucial for understanding the scale and orientation of the ellipse. The semi-major axis is determined in relation to the semi-minor axis and the eccentricity.Knowing the value of the semi-major axis allows us to define the size of the ellipse along its longest path,and it is also used in conjunction with other parameters to verify if a point lies on the ellipse as shown in the original exercise.

To find \( a^2 \) from known parameters of eccentricity and a point on the ellipse,we apply some algebraic manipulation and solve equations,as demonstrated explicitly in the solved exercise.

Coordinate axes

The coordinate axes play an important role in the positioning and orientation of the ellipse.
When an ellipse has its axes aligned with the coordinate axes, it simplifies its equation to one centered at the origin.In Cartesian coordinates,

• the x-axis represents the semi-major axis if \( a > b \), meaning the ellipse is stretched along the horizontal path.
• the y-axis represents the semi-minor axis.

For our exercise, these axes define the ellipse's orientation and are aligned with the standard x and y geometric orientation.

This positioning and alignment with the coordinate axes allow us to express the standard form of the ellipse equation in a familiar and straightforward manner,making it much simpler to handle through steps like completing the coordinate checks needed for determining if a point lies on the ellipse.