Intersection of a straight line with parabola:

Points of Intersection of Straight Line with Parabola:        

Points of intersection of y² = 4ax and y = mx + c are given by (mx + c)² = 4ax

i.e. m²x² + 2x(mc - 2a) + c² = 0                                 ......(1)

As (1) is a quadratic equation, straight line meets the parabola in 2 points (real, coincident, or imaginary). The roots of (1) are real or imaginary according as
{2(mc - 2a)}² - 4m²c² is positive or negative, that is according as -amc +a² is positive or negative, that is according as  mc is less than or greater than a.

Note:             

• When the value of m is very small, one of the roots of equation (1) is very large; when the value of m is equal to zero, this root is infinitely large. Hence every straight line parallel to the axis of the parabola meets the curve in one point at a finite distance and in another point at an infinite distance from the vertex. It means that a line parallel to the axis of the parabola meets the parabola only in one point.

Length of the Chord:

As in the preceding article, the abscissae of points common to straight line y = mx + c  and  the parabola   y²  = 4ax  are given by  equation m² x² + ( 2mc - 4a) x + c² = 0.

If (x₁, y₁) and (x₂, y₂) are points of intersection, then

(x₁ - x₂)² = (x₁ + x₂)² - 4x₁ x₂

=  and (y₁ - y₂) = m(x₁ - x₂)

Hence, required length

= 

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