-beginarray -c-c-c-c-c-c-Whine- Column 1 - - Column 2 - - Column 3 stext equation for I -x-1- 2-y- 2- 2- frac 12 x-5 y-313- 2 - i - text intersection of -P - x- 2-y- 2-16 1- - - perpendicular tangents - - Whline- text Locus of foot of -II - 7 x- 2-16 y- 2-112 - ii - perpendicular from focus -Q - x- 2-y- 2-9 1- - - upon any tangent - - lllhlineIII - - iii - Equation of chord bisected - R - 4 x -5 y -4-0 lIV - x- 2-y- 2-8 - iv - textTangent to the curveat -S - 24-10 y-1-0 1-8-10 - - 11 hline end array Which of the following is the only correct combination-A- I-i-SB- III-iii-RC- II-iii-QD- IV-iv-R

The correct option is B (III) (iii) (R) For I (i) director circle ⇒ 12x 5y + 3 = 0 (ii) tangent at vertex ⇒ 24x 10y + 1 = 0 (S) For II (i) director circle ...

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Byju's Answer

Standard XII

Mathematics

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Question

$\begin{matrix} C o l u m n 1 & C o l u m n 2 & C o l u m n 3 (equation for (Equation of conic) & locus in Column 2) Locus of point of (I) & (x - 1)^{2} + (y_{2})^{2} = (\frac{12 x - 5 y + 3}{13})^{2} & (i) & intersection of & (P) & x^{2} + y^{2} = 16 perpendicular tangents Locus of foot of (I I) & 7 x^{2} - 16 y^{2} = 112 & (i i) & perpendicular from focus & (Q) & x^{2} + y^{2} = 9 upon any tangent (I I I) & (i i i) & Equation of chord bisected & (R) & 4 x + 5 y - 4 = 0 \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1 & a t (\frac{1}{2}, \frac{2}{5}) (I V) & x^{2} + y^{2} = 8 & (i v) & Tangent to the curveat & (S) & 24 - 10 y + 1 = 0 (8, 10) \end{matrix}$

Which of the following is the only correct combination?

(I)-(i)-(S)

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(III)-(iii)-(R)

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(IV)-(iv)-(R)

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(II)-(iii)-(Q)

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Solution

The correct option is B
(III)-(iii)-(R)

For I (i) director circle $\Rightarrow 12 x - 5 y + 3 = 0$
(ii) tangent at vertex $\Rightarrow 24 x - 10 y + 1 = 0 (S)$
For II (i) director circle $\Rightarrow x^{2} + y^{2} = 9 (Q)$
(ii) auxiliary circle $\Rightarrow x^{2} + y^{2} = 16 (P)$
For III (iii) $T = S_{1}$
4x + 5y = 4 (R)
For IV (i) director circle $\Rightarrow x^{2} + y^{2} = 16 (P)$

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Similar questions

$\begin{matrix} C o l u m n I & C o l u m n 2 & C o l u m n 3 (I) & f (x) = t a n (2 t a n^{- 1} \sqrt{\frac{\sqrt{1 + \sqrt{x}} - 1}{\sqrt{1 + \sqrt{x}} + 1}}) & (i) & \int f (x) d x = \frac{4}{3} x^{\frac{3}{4}} + C & (P) & \int_{0}^{1} f (x) d x = \frac{4}{3} (I I) & f (x) = c o t (2 t a n^{- 1} \sqrt{\frac{\sqrt{1 + \sqrt{x}} - \sqrt[4]{x}}{\sqrt{1 + \sqrt{x}} + \sqrt[4]{x}}}) & (i i) & \int f (x) d x = \frac{4}{5} x^{\frac{5}{4}} + C & (Q) & \int_{0}^{1} f (x) d x = \frac{4}{5} (I I I) & f (x) ⎛ ⎜ ⎝ \frac{1 - t a n (\frac{1}{2} s i n^{- 1} (\frac{1 - \sqrt{x}}{1 + \sqrt{x}}))}{1 + t a n (\frac{1}{2} s i n^{- 1} (\frac{1 - \sqrt{x}}{1 + \sqrt{x}}))} ⎞ ⎟ ⎠ & (i i i) & \int f (x) d x = \frac{2}{3} x^{\frac{3}{4}} + C & (R) & \int_{0}^{1} f (x) d x = \frac{2}{3} (I V) & f (x) = \sqrt{x} t a n (2 t a n^{- 1} (\frac{\sqrt{\sqrt{1 + \sqrt{x} + 1}} - \sqrt{\sqrt{1 + \sqrt{x} - 1}}}{\sqrt{\sqrt{1 + \sqrt{x} + 1}} + \sqrt{\sqrt{1 + \sqrt{x} - 1}}})) & (i v) & \int f (x) d x = \frac{2}{5} x^{\frac{5}{4}} + C & (S) & \int_{0}^{1} f (x) d x = \frac{2}{5} \end{matrix}$

Which of the following options is only correct combination?

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}

The tangent to a suitable conic (Column 1) at

(\sqrt{3}, \frac{1}{2})

is found to be

\sqrt{3} x + 2 y = 4

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

Q. 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?

A line meets x-axis and y-axis at A and B respectively and O is the origin.

$\begin{matrix} Column I & Column 2 & Column 3 Equation of AB & Area of Δ O A B (I) & Centroid Δ O A B is (1, 2) & (i) & 2 x + y = 2 & (P) & 6 sq. units (II) & Circumcenter of Δ O A B is (1, 2) & (ii) & 3 x + 4 y = 12 & (Q) & 9 sq. units (III) & Distance of the orthocentre of Δ O A B & (iii) & 2 x + y = 6 & (R) & 1 sq. units From A and B is 1 and 2 respectively (IV) & Incenter of Δ O A B is (1, 1) & (iv) & 2 x + y = 4 & (S) & 4 sq. units \end{matrix}$

Which of the following is correct combination?

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The correct option is B (III)-(iii)-(R) For I (i) director circle ⇒12x−5y+3=0 (ii) tangent at vertex ⇒24x−10y+1=0(S) For II (i) director circle ⇒x2+y2=9(Q) (ii) auxiliary circle ⇒x2+y2=16(P) For III (iii) T=S1 4x + 5y = 4 (R) For IV (i) director circle ⇒x2+y2=16(P)