Question

\(A\) line \(L\) with slope 2 cuts parabola \(y^{2}=8 x\) to form a chord. If mid- point of chord lies on \(x=4\) then equation of line \(L\) is (1) \(y=2 x-6\) (2) \(x=2 y-6\) (3) \(2 x+y=6\) (4) \(x-2 y=6\)

Short answer

y=2x-6

Step-by-step solution

Understanding the given parabola equation

The equation given is a parabola: y^{2} = 8x. This represents a parabola that opens to the right.

Using the slope of the given line

The line L has a slope of 2. Therefore, the general equation of the line can be written as y = 2x + c, where c is the y-intercept.

Finding points of intersection

To find the points of intersection, substitute y = 2x + c into the parabola's equation: (2x + c)^{2} = 8x Expanding and rearranging gives: 4x^{2} + 4cx + c^{2} = 8x 4x^{2} + 4cx + c^{2} = 8x 4x^{2} + 4cx + c^{2} - 8x = 0 4x^{2} + (4c - 8)x + c^{2} = 0

Midpoint condition

The midpoint of the chord lies on x=4. Let the intersections be x_1, x_2. Then the midpoint is (x_1 + x_2) / 2 = 4.

Use Vieta’s formulas

Using Vieta’s formulas on 4x^2 + (4c - 8)x + c^2 = 0, sum of roots x_1+x_2 is given by -b/a. Thus, x_1 + x_2 = (8 - 4c) / 4 Given midpoint is 4, (x_1 + x_2) / 2 = 4 results in (8 - 4c) / 8 = 4, solving for c: 8 - 4c = 32 -4c = 24 c = -6

Determine the line equation

Substitute back c = -6 into the line equation: y=2x+c to get: y=2x-6

Key concepts

parabola

In this exercise, we encounter the equation of a parabola: \(y^2 = 8x\). A parabola is a symmetrical, U-shaped curve, defined as the set of all points (x, y) in the plane equidistant from a fixed point called the focus and a fixed line called the directrix. The given equation represents a parabola that opens to the right because the squared term is on the y-variable.
Understanding the direction in which the parabola opens is essential when analyzing its interaction with a line.

slope of a line

The slope of a line measures its steepness and is represented by the letter 'm' in the equation of a line, \(y = mx + c\). In this problem, the slope provided is 2. This means for every unit increase in x, y increases by 2 units.
The equation of the line can hence be written as \(y = 2x + c\), where 'c' is the y-intercept, which we need to determine. Understanding the slope helps us determine the exact orientation of the line concerning the parabola.

Vieta’s formulas

Vieta’s formulas relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \(x_1 + x_2\) is given by \(-b/a\).
In our problem, after expressing the combined equation of the line and the parabola, we get: \(4x^2 + (4c - 8)x + c^2 = 0\).
Using Vieta’s formulas, \( x_1 + x_2 = (8 - 4c)/4\). Given that the midpoint of the chord is 4, we use this information to solve for 'c'.
Solving \((8 - 4c)/4 = 8\) ultimately helps us find \(c = -6\).

chord of a parabola

A chord is a straight line segment connecting two points on a curve. In this context, we are looking at the chord formed by the intersection of a line with a given parabola.
Given that the midpoint of this chord lies on \(x = 4\), it provides important information needed to find where exactly the line crosses the parabola.
By integrating all the information, including the slope and the midpoint condition, we can determine the exact equation of the line, leading us to find that the value of 'c' is -6. Substituting back into the equation gives: \(y = 2x - 6\).