Columns 1- 2 and 3 contain conics- equations of tangents to the conics and points of contact- respectively- Column 1 Column 2 Column 3- I x2 - y2 - a2 i my - m2 x - a P a-m2- 2a-m- II x2 - a2y2 - a2 ii y - mx - a-m2 - 1 Q -ma-m2-1- a-m2-1- III y2 - 4ax iii y - mx - -a2m2 - 1 R -a2m-a2m2-1- 1-a2m2-1- IV x2 - a2y2 - a2 iv y - mx - -a2m2 - 1 S -a2m-a2m2-1- -1-a2m2-1- -For a - -2- if a tangent is drawn to a suitable conic Column 1 at the point of contact -1- 1- then which of the following options is the only CORRECT combination for obtaining its equation-

For a = √(2) and point (–1, 1) on the curve is x^2 + y ^2 = a^2, equation of tangent is y = mx + a√(m^2+1) and point of contact is ( ma/√(m^2+1), a/√(m^2+1)) T ...

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Byju's Answer

Standard XII

Mathematics

Equation of Tangent in Slope Form

Columns 1, 2 ...

Question

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
$\begin{matrix} C o l u m n 1 & C o l u m n 2 & C o l u m n 3 (I) x^{2} + y^{2} = a^{2} & (i) m y = m^{2} x + a & (P) (\frac{a}{m^{2}}, \frac{2 a}{m}) (I I) x^{2} + a^{2} y^{2} = a^{2} & (i i) y = m x + a \sqrt{m^{2} + 1} & (Q) (\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}}) (I I I) y^{2} = 4 a x & (i i i) y = m x + \sqrt{a^{2} m^{2} - 1} & (R) (\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}}) (I V) x^{2} - a^{2} y^{2} = a^{2} & (i v) y = m x + \sqrt{a^{2} m^{2} + 1} & (S) (\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}}) \end{matrix}$

For $a = \sqrt{2}$ , if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(- 1, 1)$ , then which of the following options is the only CORRECT combination for obtaining its equation?

(I) (i) (P)

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(I) (i i) (Q)

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(I I) (i i) (Q)

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(I I I) (i) (P)

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Solution

The correct option is B $(I) (i i) (Q)$
For $a = \sqrt{2}$ and point $(- 1, 1)$ on the curve is $x^{2} + y^{2} = a^{2}$ , equation of tangent is $y = m x + a \sqrt{m^{2} + 1}$ and point of contact is $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$

The correct combination of curve, tangent and point of contact is given in the table below:
$\begin{matrix} (I) & (i i) & (Q) (I I) & (i v) & (R) (I I I) & (i) & (P) (I V) & (i i i) & (S) \end{matrix}$

Suggest Corrections

Similar questions

Q. Columns

1, 2

and

3

contain conics, equations of tangents to the conics and points of contact, respectively.

\begin{matrix} Column - 1 & Column - 2 & Column - 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}

For

a = \sqrt{2}

, if a tangent is drawn to a suitable conic

(Column 1)

at the point of contact

(- 1, 1)

, then which of the following options is the only

CORRECT

combination for obtaining its equation?

What is the point of contact between the hyperbola
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and
the tangent $y = m x \pm \sqrt{a^{2} m^{2} - b^{2}}$ .

What is the point of contact between the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and

in tangent $y = m x \pm \sqrt{a^{2} m^{2} - b^{2}}$ .

Q. The equation

y = m x + c

is a tangent to the

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

.Then show that

c^{2} = a^{2} m^{2} + b^{2}

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}

For

a = \sqrt{2}

, if a tangent is drawn to a suitable conic (Column 1) at the point of contact

(- 1, 1)

, then which of the following options is the only CORRECT combination for obtaining its equation?

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