Compiled by: JIM M.
ASARIA
OBJECTIVES:
(a) to find the standard equation of a circle which
satisfies the given conditions
(b) to sketch its graph and indicate the center
(c) to develop perseverance and accuracy
If the center of the circle is on the origin, then h = 0 and k = 0 .
The standard equation is x² + y² = r²
Standard equation of the circle with center C (h,k) and radius, r
is (x – h )² + (y – k )² = r²
General form of the equation of the circle is
Ax² + By² + Cx + Dy + E = 0, A≠0
A circle with center (hake) and tangent to a line, Ax+By+C = 0,
the radius r, can be computed using distance formula between a
/𝑨𝒙+𝑩𝒚+𝑪/
point and a line: r=d=
𝑨𝟐 +𝑩𝟐
The coordinates of the midpoint (𝒙𝒎 , 𝒚𝒎 ) between two points
(𝒙𝟏 , 𝒚𝟏 ) and (𝒙𝟐 , 𝒚𝟐 ) can be determined using the formula :
𝒙𝟏 + 𝒙𝟐 𝒚𝟏 + 𝒚𝟐
𝒙𝒎 = , 𝒚𝒎 =
𝟐 𝟐
Distance Formula: 𝐝 = (𝒙𝟐− 𝒙𝟏 )² + (𝒚𝟐− 𝒚𝟏 )²
Review:
Determine the coordinates of the center and
radius of the circle from the equations given:
1.) 16x² + 16y² +96x – 40y= 315
2.) x² + y² –14x + 12y= 36
3.) x² + 10x+y² – 16y– 11= 0
4.) 3x² + 3y² +4y – 7 = 0
Situational Problems Involving Circles
1.) A street with two lanes, each 10 ft wide,
goes through a semicircular tunnel with
radius 12 ft. How high is the tunnel at the
edge of each lane?
Situational Problems Involving Circles
Solution: We draw a coordinate system with
origin at the middle of the highway, as
shown. Because of the given radius, the
tunnel’s boundary is on the circle x²+y²
=122. Point P is the point on the arc just
above the edge of a lane, so its x-coordinate
is 10. We need its y-coordinate. We then
solve 102 + y2 = 122 for y > 0, giving us
y = 2 𝟏𝟏 ≈ 6.63 ft.
Situational Problems Involving Circles
Situational Problems Involving Circles
Application:
Situational Problems Involving Circles
1.) A circular play area with radius 3 m is
to be partitioned into two sections using
a straight fence as shown in the figure at
the next slide. How long should the
fence be?
Situational Problems Involving Circles
Situational Problems Involving Circles
Solution:
To determine the length of the fence, we need
to determine the coordinates of its endpoints.
From the figure given, the endpoints have x
coordinate −1 and are on the circle x²+y² = 9.
Then 1+y² = 9, or y = ±2 𝟐. Therefore,
the length of the fence is 4 𝟐 ≈ 5.66m.
Assignment:
Situational Problems Involving Circles
1.) A seismological station is located at (0; −3), 3 km away
from a straight shoreline where the x-axis runs through. The
epicenter of an earthquake was determined to be 6 km away
from the station.
(a) Find the equation of the curve that contains the possible
location of the epicenter.
(b) If furthermore, the epicenter was determined to be 2 km
away from the shore, find its possible coordinates (rounded
off to two decimal places).
References:
❑Department of Education-Bureau of Learning Resources
(DEPED-BLR) (2016) Precalculus Learning Materials, Lexicon
Press Inc., Philippines.
❑Department of Education-Bureau of Learning Resources
(DEPED-BLR) (2016). Precalculus Teacher’s Guide, Lexicon
Press Inc., Philippines.
❑Larson, Ron.(2014). Precalculus with Limits 3e. Brooks/Cole,
Cengage Learning.
❑Flores, Albert & Rivera, Gemma E. (2006). Advanced Algebra
& Trigonometry. Victorious Publications Inc.
