If line 2y – x = 2 touches the parabola y2 = 2x then tangent point

1.	If line 2y – x = 2 touches the parabola y2 = 2x then tangent point is :(A) (4, 3)(B) (-4, 1)(C) (2, 2)(D) (1, 4)
Answer» Answer is (C) (2, 2) Equation of line, 2y – x = 2 x = 2y – 2 ….(i) Equation of parabola, y² = 2x From equation (i) and (ii), y² = 2(2y – 2) ⇒ y² = 4y – 4 ⇒ y² – 4y + 4 = 0 ⇒ (y – 2)² = 0 ⇒ y – 2 = 0 ⇒ y = 2 Put value of y in equation (i), x = 2 × 2 – 2 = 2 Thus tangent point, (x, y) = (2, 2)

If line 2y – x = 2 touches the parabola y2 = 2x then tangent point is :(A) (4, 3)(B) (-4, 1)(C) (2, 2)(D) (1, 4)

Answer»

Answer is (C) (2, 2)

Equation of line,

2y – x = 2

x = 2y – 2 ….(i)

Equation of parabola,

y² = 2x

From equation (i) and (ii),

y² = 2(2y – 2)

⇒ y² = 4y – 4

⇒ y² – 4y + 4 = 0

⇒ (y – 2)² = 0

⇒ y – 2 = 0

⇒ y = 2

Put value of y in equation (i),

x = 2 × 2 – 2 = 2

Thus tangent point, (x, y) = (2, 2)