Question

The position of the point (2, 1) relative to the ellipse x² + 2y² = 20 is
1. Outside the ellipse
2. Inside the ellipse but not at the focus
3. On the ellipse
4. At the focus

Answer 1

Correct Answer - Option 2 : Inside the ellipse but not at the focus

Concept:

Let general equation of curve is f(x, y) = 0

f(x, y) > 0	Point will be outside the ellipse
f(x, y) = 0	Point will be on ellipse
f(x, y) < 0	Point will be inside the ellipse

​

Calculation:

Given

Equation of ellipse, x2 + 2y2 = 20

let (x, y) = (2, 1)

⇒ f(x, y) = x² + 2y² - 20

⇒ f(2, 1) = 22 + 2 \(\rm \times \) 1² - 20 = - 14

⇒ f(2, 1) < 0

so the point will be inside the ellipse but not at the focus

 

An ellipse is formed by a plane intersecting a cone at an angle to its base. All ellipses have two focal points or foci.

The sum of the distance from every point on the ellipse to the two foci is a constant.

All ellipses have a centre and a major and minor axis.

The shape of the ellipse is in an oval shape and the area of an ellipse is defined by its major axis and minor axis.

Area of ellipse = πab, where a and b are the length of the semi-major and semi-minor axis of an ellipse.

Equation of ellipse, \(\rm \frac{x^{2}}{a^{2}}\: +\: \frac{y^{2}}{b^{2}}\: =\: 1\)

Types of ellipse

Horizontal ellipse: \(\rm \frac{x^{2}}{a^{2}}\: +\: \frac{y^{2}}{b^{2}}\: =\: 1\) , a > b

Vertical ellipse:  \(\rm \frac{x^{2}}{a^{2}}\: +\: \frac{y^{2}}{b^{2}}\: =\: 1\), a < b

 

Fundamental terms	Horizontal	Vertical
Equation of ellipse	\(\rm \frac{x^{2}}{a^{2}}\: +\: \frac{y^{2}}{b^{2}}\: =\: 1\) a > b	\(\rm \frac{x^{2}}{a^{2}}\: +\: \frac{y^{2}}{b^{2}}\: =\: 1\) a < b
Centre	(0, 0)	(0, 0)
Vertices	(\(\rm \pm \)a, 0)	(0, \(\rm \pm \)b)
Length of the major axis	2a	2b
Length of the minor axis	2b	2a
Foci	(\(\rm \pm \)ae, 0)	(0, \(\rm \pm \)be)
Equations of directrices	x = \(\rm \pm \) \(\rm \frac{a}{e}\)	y = \(\rm \pm \) \(\rm \frac{b}{e}\)
Eccentricity	\(\rm e = \sqrt{1 - \frac{b^{2}}{a^{2}}}\)	\(\rm e = \sqrt{1 - \frac{a^{2}}{b^{2}}}\)
Length of latus rectum	\(\rm \frac{2b^{2}}{a}\)	\(\rm \frac{2a^{2}}{b}\)
Distance between foci	2ae	2be
Focal distance or radii	(\(\rm \pm \)ae, \(\rm \pm \) \(\rm \frac{b^{2}}{a}\))	(\(\rm \pm \)\(\rm \frac{a^{2}}{b}\), \(\rm \pm \)be)
Sum of focal radii	2a	2b

f(x, y) > 0	Point will be inside the ellipse
f(x, y) < 0	Point will be outside the ellipse