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Column - I Column - II
A) The x-coordinate of points on the axis of the Parabola y² - 4x - 2y + 5 = 0 from which all the
three normals to the parabola are real is P) 4
B) The x-coordinate of points on the axis of the parabola 4y² -32x + 4y + 65 = 0 from which all the three normals to the parabola are real is Q) 5
C) The x-coordinate of points on the axis of the Parabola 4y² -16x - 4y + 41 = 0 from which all the three normals to the parabola are real is R) 6
D) The x-coordinate of point of the focus of the Parabola y² - 4y - 8x + 28 = 0 S) 7
A B C D
1. PS S PQRS Q
2. Q S QRS PQRS
3. PQRS S QRS Q
4. PQRS Q S Q

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answer is C.

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Detailed Solution

$A) y^{2} - 2 y = 4 x - 5, (y - 1)^{2} = 4 (x - 1)$
Condition for three normal are real is x-h > 2a
$\Rightarrow x - 1 > 2 \Rightarrow x > 3 B) 4 y^{2} - 32 x + 4 y + 65 = 0, {(y + \frac{1}{2})}^{2} = 8 (x - 2)$
For three real normals x-2 > 4, x > 6
$C) 4 y^{2} - 16 x - 4 y + 41 = 0, (y - \frac{1^{2}}{2}) = 4 (x - \frac{5}{2})$
For three real normals x-5/2>, n>4.5
$\Rightarrow y^{2} - 4 y - 8 x + 28 = 0, (y - 2)^{2} = 8 (x - 3)$
Vertex = (3, 2), a=2 $\Rightarrow$ Focus=(5, 2)

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