the number of points ab where a and b are positive integers

The number of points $$\left( {a,\,b} \right),$$ where $$a$$ and $$b$$ are positive integers lying on the hyperbola $${x^2} - {y^2} = 512$$ is :

A. $$3$$

B. $$4$$

C. $$5$$

D. $$6$$

Answer : $$4$$

Solution :
$$\eqalign{ & {a^2} - {b^2} = 512 \cr & \Rightarrow \left( {a + b} \right)\left( {a - b} \right) = {2^9} \cr & \Rightarrow \left( {a + b,\,a - b} \right) = \left( {{2^8},\,2} \right),\,\left( {{2^7},\,{2^2}} \right),\,\left( {{2^6},\,{2^3}} \right),\,\left( {{2^5},\,{2^4}} \right) \cr} $$
Since, $$a > b,\,a + b > a - b$$ therefore the other combinations like $$\left( {{2^4},\,{2^5}} \right)$$ etc cannot be accepted. $$\left( {{2^9},\,1} \right)$$ also cannot be accepted since $$a$$ and $$b$$ are positive integers.

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

View Answer

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

View Answer

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

View Answer

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

The number of points $$\left( {a,\,b} \right),$$ where $$a$$ and $$b$$ are positive integers lying on the hyperbola $${x^2} - {y^2} = 512$$ is :

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

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The number of points $$\left( {a,\,b} \right),$$ where $$a$$ and $$b$$ are positive integers lying on the hyperbola $${x^2} - {y^2} = 512$$ is :

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 $$'\alpha \,'$$

Practice More Releted MCQ Question on Hyperbola

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Releted MCQ Question on
Geometry >> Hyperbola

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Hyperbola