Question:
If the line y = mx + 1 is tangent to the parabola y2 = 4x then find the value of m.
Solution:
Given equations are,
y = mx + 1 & y2 = 4x
By solving given equations we get
(mx + 1)2 = 4x
Expanding the above equation we get
m2x2 + 2mx + 1 = 4x
On rearranging we get
m2x2 + 2mx – 4x + 1 = 0
m+x2 + x (2m – 4) + 1 = 0
As the line touches the parabola, above equation must have equal roots,
Discriminant (D) = 0
(2m – 4)2 – 4 (m2) (1) = 0
4m2 – 16m + 16 – 4m2 = 0
-16 m + 16 = 0
– m + 1 = 0
m = 1
Hence, the required value of m is 1.