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  4. If the tangents drawn at the points P and Q on the parabola y2=2 x-3 intersect at the point R (0,1), then the orthocentre of the triangle PQR is :

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Q. If the tangents drawn at the points $P$ and $Q$ on the parabola $y^2=2 x-3$ intersect at the point $R (0,1)$, then the orthocentre of the triangle $PQR$ is :

JEE MainJEE Main 2022Conic Sections

Solution:

$y^2=2 x-3$ ...(1)
Equation of chord of contact
$PQ : r =0$
$yx 1=( x +0)-3$
$y=x-3$...(2)
image
from (1) and (2)
$(x \cdot 3)^2=2 x-3 $
$ x^2-8 x+12=0$
$ (x-2)(x-6)=0 $
$ x=2 \text { or } 6$
$ y=-1 \text { or } 3$
image
$MQR =\frac{2}{6}=\frac{1}{3} $
$ MPR =\frac{2}{6}=\frac{1}{3} $
$ MPR =\frac{2}{-2}=-1 $
$ MPQ \times MPR =-\Rightarrow PQ \perp PR$
$ \text { Orthocentre }= P (2,-1)$