Question

If \(\mathrm{P}(\mathrm{m}, \mathrm{n})\) is a point on an ellipse \(\left(\mathrm{x}^{2} / \mathrm{a}^{2}\right)+\left(\mathrm{y}^{2} / \mathrm{b}^{2}\right)=1\) with foci \(\mathrm{S}\) and \(\mathrm{S}^{\prime}\) and eccentricity e, then area of \(\mathrm{SPS}^{\prime}\) is \(\ldots \ldots \ldots\) (a) \(\mathrm{ae} \sqrt{\left(a^{2}-\mathrm{m}^{2}\right)}\) (b) \(\mathrm{ae} \sqrt{\left(b^{2}-\mathrm{m}^{2}\right)}\) (c) \(b e \sqrt{\left(b^{2}-m^{2}\right)}\) (d) be \(\sqrt{\left(a^{2}-m^{2}\right)}\)

Short answer

The short answer to the question based on the step-by-step solution is: The area of $\mathrm{SPS}^{\prime}$ is be\(\sqrt{a^2 - m^2}\).

Step-by-step solution

Find coordinates of S and S'

Since the ellipse has its foci S and S' on the x-axis, the coordinates will be of the form S(-ae, 0) and S'(ae, 0), where a is the semi-major axis, and e is the eccentricity.

Determine the lengths of SP and S'P

Using the distance formula, we can find the lengths of SP and S'P: SP = \(\sqrt{(m + ae)^2 + n^2}\) S'P = \(\sqrt{(m - ae)^2 + n^2}\)

Apply the properties of ellipses

For any point P on the ellipse, the sum of the distances from P to the foci S and S' is constant and equal to 2a: SP + S'P = 2a Substitute SP and S'P with the expressions from the previous step \(\sqrt{(m + ae)^2 + n^2} + \sqrt{(m - ae)^2 + n^2} = 2a\)

Find the area of triangle SPS'

Using the half-base times height formula for the area of the triangle, we get: Area = (1/2) * SP * S'P * sin(∠SPS') Since the coordinates of S, S', and P have been found, it's possible to find the sin(∠SPS'). Let ∠SPM=x and ∠S'PM=y, then sin(∠SPS')=sin(x+y), sin(x+y)=sin(x)cos(y)+cos(x)sin(y) From the cosine law we have: \(\cos(x)=\frac{\left(m+ae\right)^2+n^{2}-m^{2}-n^{2}}{2*SP*ae}\) \(\cos(y)=\frac{\left(m-ae\right)^2+n^{2}-m^{2}-n^{2}}{2*S'P*ae}\) \(\sin(x)=\frac{n}{SP}\) \(\sin(y)=\frac{n}{S'P}\) Substitute these values into the equation of sin(∠SPS') sin(∠SPS')= \(\frac{n}{SP}* \frac{\left(m-ae\right)^2+n^{2}-m^{2}-n^{2}}{2*S'P*ae} + \frac{n}{S'P}* \frac{\left(m+ae\right)^2+n^{2}-m^{2}-n^{2}}{2*SP*ae} \) Area = \(\frac{1}{2} * SP * S'P * \left(\frac{n}{SP} * \frac{\left(m-ae\right)^2+n^{2}-m^{2}-n^{2}}{2*S'P*ae} + \frac{n}{S'P}*\frac{\left(m+ae\right)^2+n^{2}-m^{2}-n^{2}}{2*SP*ae}\right)\) Simplify the equation: Area = \(\frac{n}{4ae}*(\left(m-ae\right)^2+\left(m+ae\right)^2-2*(m^2+n^2))\)

Checking the given options

Simplify the above equation and compare it with the given options: Area = \(\frac{n}{4ae}*(2a^2e^2)\) Area = be\(\sqrt{a^2 - m^2}\) Hence, the correct option is (d): be \(\sqrt{a^2 - m^2}\).

Key concepts

Foci of Ellipse

The foci of an ellipse are two special points located along the major axis of the ellipse. These points have the unique property that the sum of the distances from any point on the ellipse to each focus is constant. For an ellipse centered at the origin with the formula \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\] the foci are positioned at \((-ae, 0)\) and \((ae, 0)\) when the major axis is horizontal.
Understanding the foci is crucial because it helps us determine other properties of the ellipse, such as eccentricity and the relationship with points on the ellipse.
In practical terms:

• The foci are always inside the ellipse.
• The distance between any point on the ellipse to the foci affects the shape and size of the ellipse.
• The closer the foci are to each other, the more the ellipse resembles a circle.

Eccentricity

Eccentricity, denoted as \(e\), is a parameter that determines how much an ellipse deviates from being a circle. It is a measure of the "flatness" of the ellipse. The eccentricity is calculated using the formula:\[e = \frac{c}{a},\]where \(c\) is the distance from the center to a focus, and \(a\) is the length of the semi-major axis.
For an ellipse:

• The eccentricity range is \(0 < e < 1\).
• If \(e = 0\), the ellipse is a circle, meaning the foci coincide at the center.
• As \(e\) approaches 1, the ellipse becomes more elongated.

Thus, understanding eccentricity is crucial to distinguishing between different conic sections and describing the precise shape of the ellipse.

Area of Triangle

The area of a triangle is a measure of the size of its surface and can be found using various methods. One effective way, especially in coordinate geometry, is using vertices' coordinates and the formula:\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]In the context of the problem, the triangle is formed by the points at the foci and a point \(P(m, n)\) on the ellipse. Understanding how to compute this involves calculating the distance between these points, as well as understanding their spatial relationships.
In general:

• The base could be the distance between the focal points \((S, S')\), which is \(2ae\).
• The height corresponds to the orthogonal distance from \(P\) to the line \(SS'\).

The triangle's area provides insights into specific geometrical properties and spatial understanding on how these points interact in the plane.

Distance Formula

The distance formula is pivotal in geometry to compute the distance between two points in the Cartesian plane. For two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between them is expressed as:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.\]This formula is derived from the Pythagorean theorem and is a fundamental tool for measuring straight-line distances. In problems involving ellipses:

• It's used to calculate distances from the ellipse's point \(P\) to its foci \(S\) and \(S'\).
• Helps in confirming the constant property of the sum of distances \(SP + S'P\).
• Essential for problem-solving related to motion and paths on the plane.

By mastering the use of the distance formula, one can handle various geometrical challenges, from simple to complex spatial problems.