If the tangents drawn at the points P and Q on the parabola y^2=2x-3 intersect at the poin…

Question

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

Accepted Answer

Chord of contact from $R(0,1)$ for $y^2-2x+3=0$: $y(1)-(x+0)+3=0\Rightarrow x-y=3$.

Intersecting with $y^2=2x-3$: $y^2=2(y+3)-3=2y+3\Rightarrow y^2-2y-3=0\Rightarrow y=3,-1$, giving $P=(6,3)$, $Q=(2,-1)$.

Slope $PQ=1$, slope $QR=\dfrac{1-(-1)}{0-2}=-1$ --- perpendicular, so $\angle PQR=90^\circ$ and the orthocentre is $Q=(2,-1)$.

Solutions · 1