  Question

# Which of the following statements is/are correct?2x+y+12a=0 is normal to the parabola y2=4ax from the point (5a,2a)x−y−3a=0 is normal to the parabola y2=4ax from the point (5a,2a)If α is the angle between two tangents drawn from the point (−2,−1) to the parabola y2=4x, then tanα=3If 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 x−y−3a=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=3Equation 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 ⇒t3−3t−2=0 (t−2)(t+1)2=0 t=2,−1,−1 Equation of normals for t=2, t=−1 are y=−2x+12a⇒2x+y−12a=0 and y=x−3a⇒x−y−3a=0 respectively. Equation of pair of tangents from (−2,−1) is given by SS1=T2 (y2−4x)(1+8)=[y(−1)−2(x−2)]2 9y2−36x=[−y−2x+4]2 9y2−36x=y2+4x2+16+4xy−16x−8y ⇒4x2−8y2+4xy+20x−8y+16=0 tanα=∣∣∣2√h2−aba+b∣∣∣ tanα=∣∣ ∣∣2√4−4(−8)4−8∣∣ ∣∣ =∣∣∣12−4∣∣∣=3  Suggest corrections   