Problem 13 - A Complete Resource Book in Mathematics for JEE Main

Question

If \((5,12)\) and \((24,7)\) are the foci of an ellipse passing through the origin, then the eccentricity of the conic is (A) \(\frac{\sqrt{386}}{12}\) (B) \(\frac{\sqrt{386}}{13}\) (C) \(\frac{\sqrt{386}}{25}\) (D) \(\frac{\sqrt{386}}{38}\)

Step-by-Step Solution

Answer

The eccentricity is \(\frac{\sqrt{386}}{38}\). The answer is (D).

Step 1: Finding the Distance Between the Foci

The first step in solving this problem is to calculate the distance between the two foci. We will use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Let \((x_1, y_1) = (5, 12)\) and \((x_2, y_2) = (24, 7)\). So: \[ d = \sqrt{(24 - 5)^2 + (7 - 12)^2} = \sqrt{19^2 + (-5)^2} = \sqrt{361 + 25} = \sqrt{386}.\] This distance is \(2c\) since the foci are \((c\times2)\) apart. Thus, \(2c = \sqrt{386}\) and \(c = \frac{\sqrt{386}}{2}\).

Step 2: Finding the Major Axis Length

The ellipse passes through the origin \((0,0)\), which lies on the ellipse. The shortest total distance from any point on the ellipse to the foci is equal to the major axis \(2a\). Given that the origin \((0,0)\) is a point on the ellipse: \[\sqrt{(5 - 0)^2 + (12 - 0)^2} + \sqrt{(24 - 0)^2 + (7 - 0)^2} = 2a.\]Evaluating the left-hand side:\[\sqrt{5^2 + 12^2} + \sqrt{24^2 + 7^2} = \sqrt{25 + 144} + \sqrt{576 + 49} = 13 + 25 = 38.\]Thus, \(2a = 38\) and \(a = 19\).

Step 3: Calculating the Eccentricity

The eccentricity \(e\) of an ellipse is given by the formula \(e = \frac{c}{a}\). We have found \(c = \frac{\sqrt{386}}{2}\) and \(a = 19\). Therefore, the eccentricity \(e\) is\[e = \frac{c}{a} = \frac{\frac{\sqrt{386}}{2}}{19} = \frac{\sqrt{386}}{38}.\]

Step 4: Determine the Correct Answer Choice

Matching the calculated eccentricity, \(\frac{\sqrt{386}}{38}\), with the given options, the correct choice is:(D) \(\frac{\sqrt{386}}{38}\).

Key Concepts

Distance Formula

The distance formula plays a crucial role in geometry, especially when dealing with problems involving distance between two points. In a 2D plane, it helps us find the exact distance between two points \((x_1, y_1)\) and \((x_2, y_2)\). The formula is derived from the Pythagorean Theorem and is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \.\] This formula essentially calculates the length of the hypotenuse (distance) in a right triangle formed by these two points. In the context of ellipses, the distance formula helps determine the distance between the foci, which is vital to understand the properties and dimensions of the ellipse.

Ellipse Foci

The foci of an ellipse are two fixed points situated along the major axis of the ellipse. These points are significant because the sum of the distances from any point on the ellipse to each foci is constant. This constant sum determines the size and shape of the ellipse. For the given problem, the foci are located at \((5, 12)\) and \((24, 7)\). Knowing these points allows us to use the distance formula to find the distance between them. This step is crucial because it eventually leads us to find important properties like the major axis and the eccentricity. Remember, in an ellipse, the two foci are symmetrically situated about the center of the ellipse.

Ellipse Equation

An ellipse is defined geometrically as the set of all points in a plane, the sum of whose distances from two fixed points, the foci, is a constant. The standard form of an ellipse equation centered at the origin \(0,0\) is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \.\] Here, \('a'\) represents the semi-major axis and \('b'\) represents the semi-minor axis. The values of \(a\) and \(b\) determine the ellipse's dimensions. If \(a > b\), the ellipse is stretched along the x-axis, meaning it's wider than tall, whereas, if \(b > a\), it's stretched along the y-axis, meaning it's taller than wide. Understanding this equation helps comprehend how changes in the foci or axis alter the overall shape of the ellipse.

Conic Sections

Ellipses are a type of conic section, which are the curves obtained by intersecting a plane with a double-napped cone. Different angles and positions of the intersecting plane result in different shapes: circles, ellipses, parabolas, and hyperbolas.

• A circle is formed when the plane is perpendicular to the cone's axis.
• An ellipse emerges when the plane cuts through the cone at an angle, but does not intersect the base.
• A parabola results when the plane is parallel to the cone's edge.
• A hyperbola occurs when the plane cuts through both nappes of the cone.

Understanding conic sections is fundamental due to the role they play in various fields such as physics, astronomy, and engineering. They describe a variety of natural phenomena and are foundational shapes in mathematics.