Question

Show that the line Y=mX+c touches the parabola Y²=4aX if c=a/m

Answer 1

Answer:

c = a/m

Step-by-step explanation:

Hi,

Given equation of the line y = mx + c

Given equation of parabola , y² = 4ax

Given that line touches the parabola, it means the line is a

tangent touching the parabola at only one point.

Hence, the points of intersection of the line with the parabola

should  result in only 1 distinct root.

Finding the points of intersection of line y = mx + c with y² = 4ax

(mx + c)² = 4ax

m²x² + (2cm - 4a)x + c² = o

which is a quadratic in x, but since the given line is a tangent

line touching at only one point, the above equation should have

equal roots, hence Discriminant = 0

(2mc - 4a)² = 4m²c²

On simplifying we get

4m²c² - 16amc + 16a² = 4m²c²

a² = amc

a(a - mc) = 0

Since a cannot be 0,

a - mc = 0

c = a/m

Hence, the line Y=mX+c touches the parabola Y²=4aX if c=a/m

Hope, it helps !