Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

IBPS Clerk Prelims - Study Material, Mock Tests

UPSC : Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

The document Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev is a part of the UPSC Course IBPS Clerk Prelims - Study Material, Mock Tests.
All you need of UPSC at this link: UPSC

1. Standard form of Circle
Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

  • (x - h)2 + (y - k)2 = r2
    Where the centre is (h, k) and radius is ‘r’.
  • If centre of the circle is at the origin and radius is 'r', then the equation of the circle is:
    x2 + y2 = r2
  • This is also known as the simplest form.

Example.1 If the area of the circle shown below is kπ, what is the value of k?
(a) 4 
(b) 16
(c) 32 
(d) 20

Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Answer. (c) 32
Solution. Since the line segment joining (4, 4) and (0, 0) is a radius of the circle:
r2 = 42 + 42 = 32
Therefore, area = πr2 = 32π ⇒ k = 32 

Try yourself:What is the coordinates of the centre and the radius of the circle with equation: 
(x - 4)2 + (y - 3)2 = 25 
View Solution

2. The Ellipse

  • Though not so simple as the circle, the ellipse is nevertheless the curve most often "seen" in everyday life. The reason is that every circle, viewed obliquely, appears elliptical.
    Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRevEllipse
  • The equation of an ellipse centered at the origin and with an axial intersection at (±, a, 0) and (± b, 0) is:
    Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

3. The Parabola

  • When a baseball is hit into the air, it follows a parabolic path. There are all kinds of parabolas, and there’s no simple, general parabola formula for you to memorize.
    Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRevParabola
  • You should know, however, that the graph of the general quadratic equation:
    y = ax2 + bx + c is a parabola. 
  • It’s one that opens up either on top or on the bottom, with an axis of symmetry parallel to the y-axis. The graph of the general quadratic equation y = ax2 + bx + c is a parabola.
    Examples: y = x2 - 2x + 1 and y = - x2 - 4 are examples of some parabolic equations.

4. The Hyperbola

  • If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed.
    Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRevHyperbola
  • The equation of a hyperbola at the origin and with foci on the x-axis is:
    Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Example.2 Find the area enclosed by the figure | x | + | y | = 4.
Solution. The four possible lines are:
x + y = 4; x - y = 4; - x - y = 4 and -x + y = 4
The four lines can be represented on the coordinates axes as shown in the figure. Thus a square is formed with the vertices as shown. The side of the square is: Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev
The area of the square is Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev= 32 sq. units.

Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Example.3 If point (t, 1) lies inside circle x2 + y2  = 10, then t must lie between:
Solution. As (t, 1) lies inside the circle, so its distance from centre i.e. origin should be less than radius i.e.Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev
Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Example.4 Find the equation of line passing through (2, 4) and through the intersection of line 4x - 3y - 21 = 0 and 3x - y - 12 = 0?
Solution. 
4x - 3y - 21 = 0 …..(1)
3x - y - 12 = 0 ….(2)
Solving (1) and (2), we get point of intersection as x = 3 & y = - 3.
Now we have two points (3, -3) & (2, 4)
⇒ Slope of line m = Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev = - 7
So, the equation of line is:
⇒ y + 3 = - 7 (x - 3)        
⇒ 7x + y - 18 = 0
Alternate Method:
Equation of line through intersection of 4x - 3y - 21 = 0 and 3x - y - 12 = 0 is:
(4x - 3y - 21) + k(3x - y - 12) = 0.
As this line passes through (2, 4):
⇒ (4 × 2 - 3 × 4 - 21) + k(3 × 2 - 4 - 12) = 0  
⇒ k = Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev
So, the equation of line is:

Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Related Searches

Summary

,

Extra Questions

,

Previous Year Questions with Solutions

,

Viva Questions

,

Semester Notes

,

Exam

,

study material

,

mock tests for examination

,

Important questions

,

Geometry

,

past year papers

,

Quantitative Reasoning Quant Notes | EduRev

,

Free

,

Ellipse and Parabola - Examples (with Solutions)

,

shortcuts and tricks

,

Sample Paper

,

Geometry

,

pdf

,

Quantitative Reasoning Quant Notes | EduRev

,

video lectures

,

Geometry

,

Ellipse and Parabola - Examples (with Solutions)

,

Circle

,

Ellipse and Parabola - Examples (with Solutions)

,

Circle

,

ppt

,

Objective type Questions

,

MCQs

,

Circle

,

Quantitative Reasoning Quant Notes | EduRev

,

practice quizzes

;