Question

Match each entry in List-I to the correct entry in List-II and choose the correct option.

• A. \(P \to (3),\ Q \to (4),\ R \to (1),\ S \to (2)\)
• B. \(P \to (3),\ Q \to (2),\ R \to (1),\ S \to (5)\)
• C. \(P \to (3),\ Q \to (2),\ R \to (4),\ S \to (5)\)
• D. \(P \to (4),\ Q \to (1),\ R \to (2),\ S \to (3)\)

Hint:

For hyperbola: \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] we have: \[ c^2=a^2+b^2 \] and directrices: \[ x=\pm\frac ae \]

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Standard equations of tangents and normals to conics are used. Distance from center to tangent for circles is the radius.
Step 3: Detailed Explanation:
1) Entry (P): Radius \( r = \frac{|3(1)+4(2)-1|}{5} = 2 \).
Eq: \( (x-1)^2 + (y-2)^2 = 4 \).
Check point (3, 2): \( (3-1)^2 + (2-2)^2 = 4 + 0 = 4 \). Correct. (P) \(\to\) (3).
2) Entry (Q): Parabola tangent: \( y = mx + 2/m \).
Distance from (0,0) to \( mx-y+2/m=0 \) is \( \sqrt{2} \).
\( \frac{2/m}{\sqrt{m^2+1}} = \sqrt{2} \implies 2 = m^2(m^2+1) \implies m^2=1 \).
Positive slope \( m=1 \). Tangent: \( y=x+2 \).
Check point (7, 9): \( 9 = 7+2 \). Correct. (Q) \(\to\) (2).
3) Entry (R): Ellipse \( x^2/16 + y^2/12 = 1 \). \( a=4, b=2\sqrt{3}, e=1/2 \).
\( M(ae, b^2/a) = (2, 3) \).
Normal at (2,3): \( \frac{a^2x}{x_1} - \frac{b^2y}{y_1} = a^2-b^2 \implies \frac{16x}{2} - \frac{12y}{3} = 4 \implies 2x - y = 1 \).
Check point (1, 1): \( 2(1)-1=1 \). Correct. (R) \(\to\) (1).
4) Entry (S): Focus \( ae=5 \), Directrix \( a/e = 16/5 \).
\( a^2 = 16 \implies a=4, e=5/4 \). \( b^2 = 16(25/16-1) = 9 \).
Hyperbola: \( x^2/16 - y^2/9 = 1 \).
Check point \( (8, 3\sqrt{3}) \): \( 64/16 - 27/9 = 4 - 3 = 1 \). Correct. (S) \(\to\) (5).
Step 4: Final Answer:
Matches (B).