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Question

Which of the following statements is/are correct?
  1. 2x+y+12a=0 is normal to the parabola y2=4ax from the point (5a,2a)
  2. xy3a=0 is normal to the parabola y2=4ax from the point (5a,2a)
  3. If α is the angle between two tangents drawn from the point (2,1) to the parabola y2=4x, then tanα=3
  4. If α is the angle between two tangents drawn from the point (2,1) to the parabola y2=4x, then tanα=13


Solution

The correct options are
B xy3a=0 is normal to the parabola y2=4ax from the point (5a,2a)
C If α is the angle between two tangents drawn from the point (2,1) to the parabola y2=4x, then tanα=3
Equation of normal to the parabola y2=4ax at the point (at2,2at) is tx+y=2at+at3
As this normal passes through (5a,2a),
5at+2a=2at+at3
t33t2=0
(t2)(t+1)2=0
t=2,1,1
Equation of normals for t=2, t=1 are y=2x+12a2x+y12a=0 and y=x3axy3a=0 respectively.


Equation of pair of tangents from (2,1) is given by SS1=T2
(y24x)(1+8)=[y(1)2(x2)]2
9y236x=[y2x+4]2
9y236x=y2+4x2+16+4xy16x8y
4x28y2+4xy+20x8y+16=0
tanα=2h2aba+b
tanα=∣ ∣244(8)48∣ ∣
=124=3

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