Consider any point P on the ellipse \(\frac{{{{\rm{x}}^2}}}{{25}} + \frac{{{{\rm{y}}^2}}}{9} = 1\) in the first quadrant. Let r and s represent its distance from (4, 0) and (-4, 0) respectively, then (r + s) is equal to
10 unit
Understanding the Ellipse and its Properties
The given question asks about a point P on the ellipse \(\frac{{{{\rm{x}}^2}}}{{25}} + \frac{{{{\rm{y}}^2}}}{9} = 1\) and the sum of its distances from two specific points, (4, 0) and (-4, 0). To solve this, we need to understand the standard form and key properties of an ellipse.
Analyzing the Ellipse Equation
The equation of the ellipse is given as \(\frac{{{{\rm{x}}^2}}}{{25}} + \frac{{{{\rm{y}}^2}}}{9} = 1\). This is in the standard form of an ellipse centered at the origin:
\[ \frac{{{{\rm{x}}^2}}}{{{{\rm{a}}^2}}} + \frac{{{{\rm{y}}^2}}}{{{{\rm{b}}^2}}} = 1 \]
By comparing the given equation with the standard form, we can identify the values of \(a^2\) and \(b^2\):
- \(a^2 = 25\)
- \(b^2 = 9\)
Taking the square root, we get:
- \(a = \sqrt{25} = 5\)
- \(b = \sqrt{9} = 3\)
Since \(a^2 > b^2\) (or \(a > b\)), the major axis of the ellipse lies along the x-axis. The length of the semi-major axis is \(a=5\), and the length of the semi-minor axis is \(b=3\).
Identifying the Foci of the Ellipse
The foci of an ellipse with its major axis along the x-axis are located at \((\pm c, 0)\), where \(c\) is calculated using the relationship \(c^2 = a^2 - b^2\). Let's calculate \(c\) for our ellipse:
\[ c^2 = a^2 - b^2 = 25 - 9 = 16 \]
Taking the square root of \(c^2\), we get:
\[ c = \sqrt{16} = 4 \]
So, the foci of the given ellipse are located at \((\pm 4, 0)\), which are (4, 0) and (-4, 0).
Applying the Ellipse Distance Property
The question states that r is the distance from the point P on the ellipse to (4, 0), and s is the distance from P to (-4, 0). We just identified that (4, 0) and (-4, 0) are the foci of the ellipse.
A fundamental property of an ellipse is that for any point P on the ellipse, the sum of its distances from the two foci is constant and equal to the length of the major axis.
Let \(F_1\) and \(F_2\) be the two foci. For any point P on the ellipse, the property states:
\[ PF_1 + PF_2 = 2a \]
In this problem, the distances r and s are precisely \(PF_1\) and \(PF_2\) (the order doesn't matter for the sum). We found that the semi-major axis length is \(a=5\).
Therefore, the sum of the distances r and s is:
\[ r + s = 2a = 2 \times 5 = 10 \]
The sum of the distances from any point on the ellipse to the foci is 10 units. The fact that point P is in the first quadrant does not change this property; it holds for any point on the ellipse.
Calculation Summary
| Parameter | Value | Source |
|---|---|---|
| Ellipse Equation | \(\frac{{{{\rm{x}}^2}}}{{25}} + \frac{{{{\rm{y}}^2}}}{9} = 1\) | Given |
| \(a^2\) | 25 | From equation |
| \(b^2\) | 9 | From equation |
| Semi-major axis \(a\) | 5 | \(\sqrt{25}\) |
| Semi-minor axis \(b\) | 3 | \(\sqrt{9}\) |
| \(c^2\) (for foci) | \(a^2 - b^2 = 25 - 9 = 16\) | Relationship |
| \(c\) | 4 | \(\sqrt{16}\) |
| Foci locations | \((\pm c, 0) = (\pm 4, 0)\) | Calculated |
| r | Distance from P to (4, 0) | Given |
| s | Distance from P to (-4, 0) | Given |
| r + s | Sum of distances from P to foci | By definition |
| Sum of distances (r + s) | 2a | Ellipse property |
| Final value of r + s | \(2 \times 5 = 10\) | Calculation |
The sum (r + s) is equal to 10 units.
Revision Table: Ellipse Properties
| Property | Description | Relevance to Problem |
|---|---|---|
| Standard Equation (center at origin, major axis on x-axis) | \(\frac{{{{\rm{x}}^2}}}{{{{\rm{a}}^2}}} + \frac{{{{\rm{y}}^2}}}{{{{\rm{b}}^2}}} = 1\) where \(a > b\) | Matches the given ellipse equation form. Helps identify a and b. |
| Foci location | \((\pm c, 0)\) where \(c^2 = a^2 - b^2\) | Helps verify if (4, 0) and (-4, 0) are the foci. |
| Sum of distances from foci | For any point P on the ellipse, the sum of distances from P to the two foci \(F_1\) and \(F_2\) is constant and equal to \(2a\) (the length of the major axis). \(PF_1 + PF_2 = 2a\) | Directly provides the value of r + s. |
| Length of Major Axis | 2a | The constant sum of distances is equal to the length of the major axis. |
Additional Information: Conic Sections
The ellipse is one of the conic sections, which are curves formed by the intersection of a plane and a double-napped cone. Other conic sections include the circle, parabola, and hyperbola. Each conic section can also be defined based on a locus of points satisfying certain distance properties related to a fixed point (focus) and a fixed line (directrix).
For an ellipse, the sum of the distances from any point on the curve to two fixed points (the foci) is constant. This is the defining property used in this problem. This property is often used to draw an ellipse using two pins and a piece of string.
The eccentricity of an ellipse, denoted by e, is defined as \(e = c/a\). For an ellipse, \(0 < e < 1\). The eccentricity measures how 'squashed' the ellipse is; an eccentricity close to 0 means it's close to a circle, while an eccentricity close to 1 means it's very elongated.
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