Problem 80 - A Complete Resource Book in Mathematics for JEE Main

Question

The radius of the circle passing through the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) and having its centre at \((0,3)\), is: (A) 4 unit (B) 3 unit (C) \(\sqrt{12}\) unit (D) \(\frac{7}{2}\) unit

Step-by-Step Solution

Answer

The radius of the circle is 4 units (option A).

Step 1: Recall the Formula for the Foci of an Ellipse

For an ellipse given by \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where \(a > b\), the foci are located at \((\pm c, 0)\) where \(c = \sqrt{a^{2} - b^{2}}\). In this problem, \(a^{2} = 16\) and \(b^{2} = 9\).

Step 2: Calculate the Distance to the Foci

First, calculate \(c\) using the formula: \[c = \sqrt{a^{2} - b^{2}} = \sqrt{16 - 9} = \sqrt{7} \] Thus, the foci are at \((\pm \sqrt{7}, 0)\).

Step 3: Determine the Centre of the Circle

The center of the circle is given as \((0,3)\).

Step 4: Calculate the Radius of the Circle

The radius of the circle is the distance from the center of the circle \((0,3)\) to one of the foci \((\sqrt{7}, 0)\). Use the distance formula: \[\text{Distance} = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}} = \sqrt{(\sqrt{7} - 0)^2 + (0 - 3)^2}\] Simplify: \[\text{Distance} = \sqrt{7 + 9} = \sqrt{16} = 4\]

Step 5: Verify with Given Options

The calculated radius of \(4\) units matches option (A).

Key Concepts

Foci of an Ellipse

Elliptical geometry can be quite fascinating! One key component of any ellipse is its foci. An ellipse has two foci, and they play a crucial role in defining the shape's unique properties. To find the foci of an ellipse given by the equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where \(a > b\), we use the formula for the location of the foci:

• \(c = \sqrt{a^{2} - b^{2}}\)
• The foci are positioned at \((\pm c, 0)\)

In our exercise, \(a^{2} = 16\) and \(b^{2} = 9\), so by substituting these values into the formula, we find \(c = \sqrt{16 - 9} = \sqrt{7}\). Hence, the foci of our ellipse are located at \((\pm \sqrt{7}, 0)\). This means the foci lie on the x-axis, at equal distances from the center of the ellipse. They essentially help in governing the 'stretch' of the ellipse, setting it apart from other geometric figures like circles.

Distance Formula

Calculating the distance between two points is a fundamental concept in geometry. It's especially useful when analyzing shapes like ellipses and circles. We use the distance formula to find the separation between two points \((x_1, y_1)\) and \((x_2, y_2)\). The formula is as follows:

• \(\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

In our problem, we want to calculate the distance between the center of a circle \((0, 3)\) and one of the foci \((\sqrt{7}, 0)\). Substituting these points into the formula, we have:
\(\text{Distance} = \sqrt{(\sqrt{7} - 0)^2 + (0 - 3)^2} = \sqrt{7 + 9} = \sqrt{16} = 4\).
The solution tells us that the radius of our concerned circle is 4 units, as this is the distance between these two pivotal points.

Circle Radius Calculation

Understanding radius calculation is central to solving this type of geometry problem. A circle's radius is defined as the distance from the center to any point on its perimeter. In our scenario, this circle passes through the foci of the ellipse, and its center is given as \((0, 3)\).
To find the radius, we already calculated the distance from the center of the circle to one of the ellipse's foci using the distance formula. We found this distance to be 4 units, thus making it the circle's radius.

• Important: The radius was calculated between the circle's center \((0, 3)\) and a focus \((\sqrt{7}, 0)\).
• This demonstrates how circles and ellipses are often interconnected via their geometric properties.

The exercise's goal, calculating the circle radius, thereby demonstrates the blend of different geometrical concepts like ellipses and circles, showcasing how concepts overlap and work together.