if the line y mx a 2m 2 b 2 touches the hyperbola x 2a 2 y

If the line $$y = mx + \sqrt {{a^2}{m^2} - {b^2}} $$ touches the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$$ at the point $$\varphi .$$ Then $$\varphi = \,?$$

A. $${\sin ^{ - 1}}\left( m \right)$$

B. $${\sin ^{ - 1}}\left( {\frac{a}{{bm}}} \right)$$

C. $${\sin ^{ - 1}}\left( {\frac{b}{{am}}} \right)$$

D. $${\sin ^{ - 1}}\left( {\frac{{bm}}{a}} \right)$$

Answer : $${\sin ^{ - 1}}\left( {\frac{b}{{am}}} \right)$$

Solution :
Equation of tangent at point $$'\varphi '$$ on the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$$ is
$$\eqalign{ & \frac{x}{a}\sec \,\varphi - \frac{y}{b}\tan \,\varphi = 1 \cr & {\text{or }}y = \frac{b}{a}x\,{\text{cosec}}\,\varphi - b\,\cot \,\varphi ......\left( 1 \right) \cr & {\text{If }}y = mx + \sqrt {{a^2}{m^2} - {b^2}} ......\left( 2 \right) \cr} $$
also touches the hyperbola then on comparing $$\left( 1 \right)$$ & $$\left( 2 \right)$$
$$\eqalign{ & 1 = \frac{{\frac{b}{a}{\text{cosec}}\,\varphi }}{m} = \frac{{ - b\,\cot \,\varphi }}{{\sqrt {{a^2}{m^2} - {b^2}} }} \cr & {\text{Hence, }}m = \frac{b}{a}{\text{cosec}}\,\varphi \,; \cr & {\text{or cosec}}\,\varphi = \frac{{am}}{b} \cr & {\text{or }}\sin \,\varphi = \frac{b}{{am}} \cr & {\text{or }}\varphi = {\sin ^{ - 1}}\frac{b}{{am}} \cr} $$

Releted MCQ Question on
Geometry >> Hyperbola

Releted Question 1

Each of the four inequalities given below defines a region in the $$xy$$ plane. One of these four regions does not have the following property. For any two points $$\left( {{x_1},\,{y_1}} \right)$$ and $$\left( {{x_2},\,{y_2}} \right)$$ in the the region, the point $$\left( {\frac{{{x_1} + {x_2}}}{2},\,\frac{{{y_1} + {y_2}}}{2}} \right)$$ is also in the region. The inequality defining this region is :

A. $${x^2} + 2{y^2} \leqslant 1$$

B. $${\text{max }}\left\{ {\left| x \right|,\left| y \right|} \right\} \leqslant 1$$

C. $${x^2} - {y^2} \leqslant 1$$

D. $${y^2} - {x^2} \leqslant 0$$

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Releted Question 2

Let $$P\left( {a\,\sec \,\theta ,\,b\,\tan \,\theta } \right)$$ and $$Q\left( {a\,\sec \,\phi ,\,b\,\tan \,\phi } \right),$$ where $$\theta + \phi = \frac{\pi }{2},$$ be two points on the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1.$$ If $$\left( {h,\,k} \right)$$ is the point of intersection of the normal at $$P$$ and $$Q,$$ then $$k$$ is equal to :

A. $$\frac{{{a^2} + {b^2}}}{a}$$

B. $$ - \left( {\frac{{{a^2} + {b^2}}}{a}} \right)$$

C. $$\frac{{{a^2} + {b^2}}}{b}$$

D. $$ - \left( {\frac{{{a^2} + {b^2}}}{b}} \right)$$

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Releted Question 3

If $$x=9$$ is the chord of contact of the hyperbola $${x^2} - {y^2} = 9,$$ then the equation of the corresponding pair of tangents is :

A. $$9{x^2} - 8{y^2} + 18x - 9 = 0$$

B. $$9{x^2} - 8{y^2} - 18x + 9 = 0$$

C. $$9{x^2} - 8{y^2} - 18x - 9 = 0$$

D. $$9{x^2} - 8{y^2} + 18x + 9 = 0$$

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Releted Question 4

For hyperbola $$\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1,$$ which of the following remains constant with change in $$'\alpha \,'$$

A. abscissae of vertices

B. abscissae of foci

C. eccentricity

D. directrix

View Answer

If the line $$y = mx + \sqrt {{a^2}{m^2} - {b^2}} $$ touches the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$$ at the point $$\varphi .$$ Then $$\varphi = \,?$$

Releted MCQ Question on
Geometry >> Hyperbola

If $$x=9$$ is the chord of contact of the hyperbola $${x^2} - {y^2} = 9,$$ then the equation of the corresponding pair of tangents is :

For hyperbola $$\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1,$$ which of the following remains constant with change in $$'\alpha \,'$$

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Hyperbola

Practice More MCQ Question on Maths Section

If the line $$y = mx + \sqrt {{a^2}{m^2} - {b^2}} $$ touches the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$$ at the point $$\varphi .$$ Then $$\varphi = \,?$$

Releted MCQ Question on Geometry >> Hyperbola

If $$x=9$$ is the chord of contact of the hyperbola $${x^2} - {y^2} = 9,$$ then the equation of the corresponding pair of tangents is :

For hyperbola $$\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1,$$ which of the following remains constant with change in $$'\alpha \,'$$

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Releted MCQ Question on
Geometry >> Hyperbola

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Hyperbola