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CIVIL ENGINEERING BOARD EXAMS PROBLEMS PHILIPPINES โ October 30, 2020
IDENTIFICATION OF CONIC SECTIONS AND NORMAL AND TANGENT TO CONICS
GENERAL EQUATION OF THE CONICS
IDENTIFYING CONIC SECTIONS BY ECCENTRICITY (e) :
- Circles โ e = 0
- Parabolas - e = 1
- Ellipses โ 0 < e < 1
- Hyperbolas โ e > 1
TANGENTS AND NORMAL TO CONICS
Unlike other curves, the tangent to any conic will pass only through one point. The following substitutions to be used in solving the tangents and normal to
conics:
For other functions, you may refer to differential calculus. Sub normal โ means the distance from point of tangency to x-axis.
Cases:
- Finding the equation of the tangent given the point on the conic โ replace variables in the conic and substitute for (x`1 , y 1 ) the given point.
- Finding the equation of the tangent that passes through a given point outside the conic โ apply necessary replacements of variables leaving x 1
and
y 1
unknown. Another equation relating x 1
and y 1
can be found by substituting (x 1
, y 1
) to the equation of the conic. By expressing y 1
in terms of x 1
in
either equation and substituting the other equation, a quadratic equation is derived in the form Ax 1 + Bx 1 + C = 0. With (x1, y 1 ) known, the tangent is
solvable.
- Finding the equation of the tangent given the slope m of the tangent โ use slope intercept form for the tangent line with m and b unknown. Since
the line and the conic crosses, substitute this value of y to the y value in the conics resulting to quadratic equation. Since tangent passes through
one distinct point, solve using the discriminant formula. With b known and m given, the tangent can now be solved.
Note: For the steps on the other cases in non-conic sections, refer to differential calculus.
EXAMPLES
- Identify the conic section 6y
2
SOLUTION:
- Identify the conic section 2x
2
2
SOLUTION:
- Find an equation of the circle with center (1, - 5) and tangent to the line 3x + 4y = 8.
SOLUTION:
DIAMETER OF CONICS
Diameter โ locus of the midpoints of a system of parallel chords.
PARABOLA
ELLIPSES
HYPERBOLAS
*Conjugate diameter โ two diameters of an ellipse or hyperbola are conjugate if each conic bisects the chords parallel to the other.
Polar and pole โ If tangents AB and AC are drawn tangent to a conic, from A (x 1 , y 1 ) external to the conic , then the line through the points of tangency B and
C is called the POLAR of the point A with respect to the conic. Conversely, if a line is drawn cutting the conic B and C and tangents constructed at these points
intersects at A, then A is called the POLE of the with the respect to the conic.
EQUATIONS OF THE POLAR
A. ELLIPSE
- (CE Board) A parabola has an equation y
2
= 8x. Find the equation of the diameter of parabola which bisect chords parallel to the line x โ y = 4.
SOLUTION:
- The equation of the ellipse is given as 16x
2
2
= 576. Find the equation of the polar of the given point (4, - 6) with respect to the ellipse.
SOLUTION:
EXERCISES - Answer the following questions.
- Write the equation xy = 1 in terms of rotated system when ฮธ = 45ยฐ. Ans. xโ
2
2
- Given is the polar 4x + y + 12 = 0 of the parabola y
2
= 4x , find the coordinates of the pole. Ans. (3 , 1/2 )
- An ellipse has an equation equal to 9x
2
2
= 225. Determine the equation of the polar of the point (2,-3) with respect to the ellipse.
Ans. 6x โ 25y = 225
- Transform this conic into standard translated form: y
2
- 6y โ 4x + 1 = 0. Ans. yโ
2
= 4xโ
- Transform this conic into standard rotated form: x
2
- y = 0 when ฮธ = 60ยฐ. Ans. xโ
2
2
- Write an equation of the translated conic in general form: x
2
2
= 7 at (3,2). Ans. x
โ
2
- A parabola has an equation y
2
= 8x. Find the equation of the tangent to the parabola having a slope parallel to line x โ y = 4. Ans. x โ y + 2 = 0
- Find a new representation of the equation 2x
2
2
- 30 = 0 when it is rotated to 45ยฐ. Ans. xโ
2
/20 + yโ
2
= 1
PARAMETRIC EQUATIONS
The variable t is called a parameter and does not appear on the graph. Equations are called parametric equations because both x and y are expressed in terms
of the parameter t. Eliminating the parameter is the most common method in solving parametric equations.
EXAMPLES:
SOLUTION:
๐ก + 2 , ๐ฆ(๐ก) = log ๐ก
SOLUTION:
EXERCISES - Eliminate the parameters on the following equations. If trigonometric equations are given, use the interval between 0 and 2ฯ.
2
2
2
2
2
2
= 4 cos ๐ก , ๐ฆ
= 3 sin ๐ก ๐ด๐๐ . 9 ๐ฅ
2
2
- ๐ฅ = 3 sin ๐ก , ๐ฆ = โ 4 cos ๐ก ๐ด๐๐ .
2
2
โ๐ก
๐ก
- ๐ฅ = tan 3 ๐ก , ๐ฆ = 2 + sec 3 ๐ก ๐ป๐๐๐ก: ๐๐ ๐ ๐๐ฆโ๐ก๐๐๐๐๐๐๐ ๐๐๐๐๐ก๐๐ก๐๐๐ . ๐ด๐๐ . ๐ฅ
2
2
Next (Final) Topics on November 6, 2020:
- Polar Coordinates and Conversion of Rectangular to Polar and Vice Versa
- Solid Analytic Geometry
