The set of points on the axis of the parabola y2-2y-4x+5=0 from which all the three normals drawn to the parabola are real and distinct, is

Question

The set of points on the axis of the parabola y2-2y-4x+5=0 from which all the three normals drawn to the parabola are real and distinct, is (A)  (x , 1): x > 3  (B)  (x , - 1): x ≥ 1  (C)  (x , 3): x ≥ 1  (D)  (x , - 3): x ≥ 3

Tardigrade Admin · Accepted Answer

Given equation of the parabola can be rewritten as

(y - 1)^{2} = 4 (x - 1)

Axis of the parabola is

y = 1

,
equation of normal is

(y - 1) = m (x - 1) - 2 m - m^{3}

Let

(h, 1)

is a point on its axis, then

0 = m (h - 1) - 2 m - m^{3}

\Rightarrow m^{2} = h - 3

\Rightarrow h > 3

for real & distinct values of

m

.

Q. The set of points on the axis of the parabola $y^{2} - 2 y - 4 x + 5 = 0$ from which all the three normals drawn to the parabola are real and distinct, is

${(x, 1) : x > 3}$

${(x, - 1) : x \geq 1}$

${(x, 3) : x \geq 1}$

${(x, - 3) : x \geq 3}$

Q. The set of points on the axis of the parabola y2−2y−4x+5=0 from which all the three normals drawn to the parabola are real and distinct, is

{(x,1):x>3}

{(x,−1):x≥1}

{(x,3):x≥1}

{(x,−3):x≥3}

Given equation of the parabola can be rewritten as (y−1)2=4(x−1) Axis of the parabola is y=1 , equation of normal is (y−1)=m(x−1)−2m−m3 Let (h,1) is a point on its axis, then 0=m(h−1)−2m−m3 ⇒m2=h−3 ⇒h>3 for real & distinct values of m .

Q. The set of points on the axis of the parabola $y^{2} - 2 y - 4 x + 5 = 0$ from which all the three normals drawn to the parabola are real and distinct, is

${(x, 1) : x > 3}$

${(x, - 1) : x \geq 1}$

${(x, 3) : x \geq 1}$

${(x, - 3) : x \geq 3}$