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

IBPS Clerk Prelims - Study Material, Mock Tests

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Standard form of Circle
(x - h)2 + (y - k)2 = r2

Where centre is (h, k) and radius is ‘r’

If centre of the circle is at the origin and radius is 'r', then equation of the circle is
x2 + y2 = r2

This is also known as the simplest form.

Ex.1 If the area of the circle shown below is kπ,
 what is the value of k?
 (1) 4 

(2) 16
 (3) 32 

(4) 20

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


Sol. 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. Answer: (3)

 


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.

The equation of an ellipse centered at the origin and with axial intersection at (±, a, 0) and (± b,0) is:

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


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. 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 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. Here, for example are parabolas representing the equations; 

y = x2 - 2x + 1 and y = - x2 = 4.


The Hyperbola

If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed.
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


Ex.2 Find the area enclosed by the figure | x | + | y | = 4.
 

Sol. 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



Ex.3 If point (t, 1) lies inside circle x2 + y2  = 10, then t must lie between

Sol. As (t, 1) lies inside 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 | EduRevCircle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev


Ex.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?

 

Sol. 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 equation of line y + 3 = - 7 (x - 3)        or         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 or k = Circle, Ellipse and Parabola - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

So, equation of line is

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

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