Let P be a variable point on the parabola y = 4 x 2 + 1 . Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is:

Step-by-step solution

We have, y = 4 x 2 + 1 L : y = x Let the foot of perpendicular from P to line y = x is Q . Let P ≡ x , y , Q ≡ c , c and R ≡ h , k where, R is the mid-point of P Q . Clearly, P Q ⊥ L ⇒ k - c h - c = - 1 ⇒ c = h + k 2 And, R ≡ x + c 2 , y + c 2 ⇒ R ≡ x 2 + h 4 + k 4 , y 2 + h 4 + k 4 Hence, h = x 2 + h 4 + k 4 ⇒ x = 3 h 2 - k 2 k = y 2 + h 4 + k 4 ⇒ y = 3 k 2 - h 2 Now, y = 4 x 2 + 1 ⇒ 3 k - h 2 = 4 3 h - k 2 2 + 1 ⇒ 3 k - h = 2 3 h - k 2