Match the following: (3, 0) is the point from which three normals are drawn to the parabola y 2 = 4 x which meet the parabola in the points P, Q and R. Then [2006]

Question

Match the following: (3, 0) is the point from which three normals are drawn to the parabola $y^{2} = 4 x$ which meet the parabola in the points P, Q and R. Then [2006]

	Column I		Column II
(A)	Area of $∆ P Q R$	(p)	2
(B)	Radius of circumcircle of $∆ P Q R$	(q)	$\frac{5}{2}$
(C)	Centroid of $∆ P Q R$	(r)	$(\frac{5}{2}, 0)$
(D)	Circumcentre of $∆ P Q R$	(s)	$(\frac{2}{3}, 0)$

Smart Achievers · Accepted Answer

(4)

$Let y = m x - 2 m - m^{3} be the equation of normal to y^{2} = 4 x .$

$∆ P Q R$ $∴$ $∆ P Q R$

$Hence three points on parabola are given by (m^{2}, - 2 m) for m = 0, 1, - 1$

$∴$ $P (0, 0), Q (1, 2) and R (1, - 2)$

$∴$ $Area (∆ P Q R) = \frac{1}{2}$ $| \begin{matrix} 0 & 0 & 1 \\ 1 & 2 & 1 \\ 1 & - 2 & 1 \end{matrix} |$ $= 2$

$Radius of circumcircle, R = \frac{a b c}{4 Δ} = \frac{\sqrt{5} \times \sqrt{5} \times 4}{4 \times 2} = \frac{5}{2}$

$(where, a, b, c are the sides of ∆ P Q R)$

$Centroid of ∆ P Q R = (\frac{2}{3}, 0); Circumcentre = (\frac{5}{2}, 0)$

Solution

1 $(A) \to (q); (B) \to (p); (C) \to (r); (D) \to (s)$

2 $(A) \to (r); (B) \to (s); (C) \to (q); (D) \to (p)$

3 $(A) \to (r); (B) \to (q); (C) \to (s); (D) \to (p)$

4 $(A) \to (p); (B) \to (q); (C) \to (s); (D) \to (r)$

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Solution

1 (A)→(q); (B)→(p); (C)→(r); (D)→(s)

2 (A)→(r); (B)→(s); (C)→(q); (D)→(p)

3 (A)→(r); (B)→(q); (C)→(s); (D)→(p)

4 (A)→(p); (B)→(q); (C)→(s); (D)→(r)

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1 $(A) \to (q); (B) \to (p); (C) \to (r); (D) \to (s)$

2 $(A) \to (r); (B) \to (s); (C) \to (q); (D) \to (p)$

3 $(A) \to (r); (B) \to (q); (C) \to (s); (D) \to (p)$

4 $(A) \to (p); (B) \to (q); (C) \to (s); (D) \to (r)$