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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Parabola

NTA Abhyas 2020 · Mathematics · Parabola

The set of points on the axis of the parabola y^ 2 - 2y - 4x + 5 =...

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

Options

  1. A(x,1):x>3{ \left(x, 1\right): x > 3 }
  2. B(x,−1):x≥1{ \left(x, - 1\right): x \ge 1 }
  3. C(x,3):x≥1{ \left(x, 3\right): x \ge 1 }
  4. D(x,−3):x≥3{ \left(x, - 3\right): x \ge 3 }

Correct answer

A. (x,1):x>3{ \left(x, 1\right): x > 3 }

Step-by-step solution

Given equation of the parabola can be rewritten as

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

Axis of the parabola is y=1y = 1,
equation of normal is

\left(y - 1\right) = m\left(x - 1\right) - 2m - m^{3}<br/><br/>Let<br /><br />Let( h, 1 )isapointonitsaxis,then<br/><br/>is a point on its axis, then<br /><br />0 = m\left(h - 1\right) - 2m - m^{3}<br/><br/><br /><br />⇒ m^{2} = h - 3<br/><br/><br /><br />⇒ h > 3for real & distinct values ofm
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