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Tangents drawn at the points P and Q on the parabola y= 2x − 3 intersect at the point R(0,1) then the ortho centre of the triangle PQR is

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\(y^2 = 2x - 3 \)   .....(1)

Equation of chord of contact

\(PQ: y.1 = (x + 0) - 3\)

⇒ \(y = x -3\)    .....(2)

From (1) & (2),

\((x-3)^2 = 2x - 3\)

⇒ \(x^2 - 8x+ 12 = 0\)

⇒ \((x - 2) (x - 6) =0\)

⇒ \(x = 2 \;or \;x = 6\)

⇒ \(y = -1 \;or\; y = 3\)

\(\therefore P (2, - 1) \text{ & }Q(6, 3)\)

\(M_{PQ} = \frac {3 -(-1)}{6-2} = \frac 44 = 1\)

\(M_{PR} = \frac{1-(-1)}{0-2} = -1\)

\(M_{QR} =\frac{1-3}{0-6} = \frac{-2}{-6} = \frac 13\)

\(\because M_{PQ}M_{PR} = 1 \times -1 = -1\)

\(\therefore \triangle PQR\) is right angle at P.

\(\therefore\) orthocentre is P(2, -1).

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