Column 1 contains equations of various conics for different parameters Column 2 contains information about tangents- normal- chords and points Column 3 gives the equations of these tangents and normal or the co-ordinate of pointsI - y- 2-4 ax - i -Have a common tangent-P-Whose equation is x - allthline II - y - 2-4 x - a - ii -Have a a common normal-Q-Whose equation is X -allthlineend array Which of the following options is the only correct combination-A- IIiiRB- IIIiRC- IViPD- IV iv S

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Direct Common Tangent

Column 1 cont...

Question

Column 1 contains equations of various conics for different parameters
Column 2 contains information about tangents, normal, chords and points
Column 3 gives the equations of these tangents and normal or the co-ordinate of points
$\begin{matrix} C o l u m n I & C o l u m n 2 & C o l u m n 3 (I) & y^{2} = 4 a x & (i) & H a v e a c o m m o n t a n g e n t & (P) & W h o s e e q u a t i o n i s x = a (I I) & y^{2} = 4 (x - a) & (i i) & H a v e a c o m m o n n o r m a l & (Q) & W h o s e e q u a t i o n i s x = - a (I I I) & \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 & (i i i) & H a v e a c o m m o n c h o r d & (R) & W h o s e e q u a t i o n i s y = 0 (I V) & x^{2} = 4 (y - a) & (i v) & H a v e a c o m m o n p o i n t & (S) & W h o s e c o - o r d i n a t e i s (a, 0) (T) & W h o s e c o - o r d i n a t e i s (- a, 0) \end{matrix}$
Which of the following options is the only correct combination?

(II)(ii)(R)

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(III)(i)(R)

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(IV)(i)(P)

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(IV)(iv)(S)

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Solution

The correct option is A (II)(ii)(R)

$y^{2} = 4 a x$
any tangent $t y = x + a t^{2}$ is not tangent to the circle
$y^{2} = 4 (x - a)$
x = a is common tangent, x - y = a is a common chord
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$
$x = \pm a$ are common tangents

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Similar questions

Q. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

\begin{matrix} C o l u m n 1 & C o l u m n 2 & C o l u m n 3 (I) x^{2} + y^{2} = a^{2} & (i) m y = m^{2} x + a & (P) (\frac{a}{m^{2}}, \frac{2 a}{m}) (I I) x^{2} + a^{2} y^{2} = a^{2} & (i i) y = m x + a \sqrt{m^{2} + 1} & (Q) (\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}}) (I I I) y^{2} = 4 a x & (i i i) y = m x + \sqrt{a^{2} m^{2} - 1} & (R) (\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}}) (I V) x^{2} - a^{2} y^{2} = a^{2} & (i v) y = m x + \sqrt{a^{2} m^{2} + 1} & (S) (\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}}) \end{matrix}

If a tangent to a suitable conic (Column 1) is found to be

y = x + 8

and its point of contact is

(8, 16)

, then which of the following options is the only CORRECT combination?

Column 1 contains the equation for 4 different functions
Column 2 contains information about increasing/decreasing nature of f(x)
Column 3 contains information about the limiting behaviour of f(x) near 0 and at infinity
$\begin{matrix} C o l u m n I & C o l u m n 2 & C o l u m n 3 (I) & f (x) = x^{2} l o g x & (i) & f (x) h a s o n e p o i n t o f m i n i m a & (P) & l i m_{x \to O^{+}} f (x) = 0 (I I) & f (x) = x l o g x & (i i) & f (x) h a s o n e p o i n t o f m a x i m a & (Q) & l i m_{x \to \infty} f (x) = 0 (I I I) & f (x) = \frac{l o g x}{x} & (i i i) & f (x) i n c r e a s e s i n (0, e) & (R) & l i m_{x \to \infty} f (x) = \infty (I V) & f (x) = x^{- x} & (i v) & f (x) d e c r e a s e s i n (0, \frac{1}{e}) & (S) & l i m_{x \to O^{+}} f (x) = 1 \end{matrix}$
Which of the following options is the only correct combination?