If the length of latusrectum of ellipse E1:...

If the length of latusrectum of ellipse $E_{1}: 4(x+y-1)^{2}+2(x-y+3)^{2}=8$
and $E_{2}: \frac{x^{2}}{p}+\frac{y^{2}}{p^{2}}=1,(0<p<1)$ are equal, then area of ellipse $E_{2}$, is
(a) $\frac{\pi}{2}$
(b) $\frac{\pi}{\sqrt{2}}$
(c) $\frac{\pi}{2 \sqrt{2}}$
(d) $\frac{\pi}{4}$

Key Concepts

Ellipse

An ellipse is a conic section defined as the locus of points such that the sum of the distances from any point on the curve to two fixed points (foci) is constant. This definition yields various geometric properties including its major and minor axes, foci, center, and eccentricity, which are essential for analyzing its shape and size.

Standard Form of Ellipse

The standard form of an ellipse equation, typically written as (x-h)²/a² + (y-k)²/b² = 1, is fundamental as it directly reveals the center (h, k), and the lengths of the semi-major axis a and the semi-minor axis b. Converting any ellipse to this form (or a related rotated version) is a key step in extracting its geometric properties and applying standard formulas.

Latus Rectum

The latus rectum of an ellipse is a line segment perpendicular to the major axis that passes through a focus, with its endpoints lying on the ellipse. Its length is given by the formula 2b²/a for an ellipse in standard form. Understanding the latus rectum is important in the study of conic sections as it relates to the focal properties and shape of the ellipse.

Area of an Ellipse

The area of an ellipse is computed using the formula ?ab, where a and b are the lengths of the semi-major and semi-minor axes, respectively. This formula encapsulates how the scaling along the two perpendicular directions affects the total size of the ellipse, and is a straightforward application once the ellipse is expressed in its standard form.

Transcript

Please give Ace some feedback