subject: Parabola And Line Intersection [print this page] A parabola is distinct as large set of points in a plane that have the same distance from a given set of point and a given line in that plane. The given point is known as Focus, and the line is called Directrix. The midpoint of right angles generates focus to the directrix is called the vertex of the parabola.
The parabola is used in many applications, like headlight reflectors to the design of ballistic missiles. They are normally used in physics, Engineering and many other areas.
Discussion on Parabola and Line Intersection
The intersection of a line is defined as the two lines are meet at a same point. In a linear equation, to find the intersection point by using two methods. (a) Substitution method (b) Elimination method
Problems on Parabola and Line Intersection:
Does the parabola y= -1/3x^2 - 7x - 15 ever intersect the line y= 3?
Solution:
Step 1: y= -1/3x^2 - 7x 15 [Given Equation]
Step 2: If y= -1/3x^2 - 7x 15 ever intersects y = 3, then we want to find out the real value to set them equal to one another.
Step 3: -1/3x^2 - 7x 15 = 3
-1/3x^2 - 7x 15 = 3 (Multiplied the whole equation with -3)
x^2 7(-3)x 15(-3) = 3(-3)
x^2 + 21x + 45 + 9 = 0
x^2 + 21x + 54 = 0
(x + 18) (x + 3) = 0
Step 4: Now equate this to zero,
x + 18 = 0 ; x + 3 = 0
x + 18 18 = -18; x + 3 3 = - 3
x = -18; x = -3
Step 5: So, it intercepts y = 3 at two places; (-18, 3) and (-3, 3)
Problem 2:
In how many points does the parabola y= 5x^2 - 3x - 7 intersect the line y= -11?
Solution:
Step 1: y= 5x^2 - 12x 7 [Given Expression]
Step 2: If y= 5x^2 - 12x 7 ever intersects y = -11, then we want to find out the points in the parabola.
Step 3: Simplify the equation,
5x^2 - 12x - 7 = -11
5x^2 - 12x 7 + 11= 0
5x^2 - 12x + 4 = 0
Step 5: now, factor the terms,
(5x - 2)(x 2) = 0
Step 6: Now, evaluate the terms
5x 2 = 0; x 2 = 0
5x 2 + 2 = + 2; x 2 + 2 = 2
5x = 2; x =2
5x / 5 = 2 /5;
x = 2 /5 and x =2
Step 7: Here it intersects twice, once at (2/5 , -11) and another at (2, -11).
