if the coordinates of four concyclic points on the

If the coordinates of four concyclic points on the rectangular hyperbola $$xy = {c^2}$$ are $$\left( {c{t_{\text{i}}},\,\frac{c}{{{t_{\text{i}}}}}} \right),\,{\text{i}} = 1,\,2,\,3,\,4$$ then :

A. $${t_1}{t_2}{t_3}{t_4} = - 1$$

B. $${t_1}{t_2}{t_3}{t_4} = 1$$

C. $${t_1}{t_3} = {t_2}{t_4}$$

D. $${t_1} + {t_2} + {t_3} + {t_4} = {c^2}$$

Answer : $${t_1}{t_2}{t_3}{t_4} = 1$$

Solution :
Let the points lie on the circle $${x^2} + {y^2} + 2gx + 2fy + 2fy + k = 0,$$ then
$$\eqalign{ & {c^2}t_{\text{i}}^2 + \frac{{{c^2}}}{{t_{\text{i}}^2}} + 2gc{t_{\text{i}}} + 2f\frac{c}{{{t_{\text{i}}}}} + k = 0 \cr & \Rightarrow {c^2}t_{\text{i}}^4 + 2gct_{\text{i}}^3 + kt_{\text{i}}^3 + 2fc{t_{\text{i}}} + {c^2} = 0 \cr} $$
Its roots are $${t_1},\,{t_2},\,{t_3},\,{t_4}$$ so $${t_1}{t_2}{t_3}{t_4} = \frac{{{c^2}}}{{{c^2}}} = 1$$
Also, $${t_1} + {t_2} + {t_3} + {t_4} = - \frac{{2gc}}{{{c^2}}} = - \frac{{2g}}{c}$$

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

If the coordinates of four concyclic points on the rectangular hyperbola $$xy = {c^2}$$ are $$\left( {c{t_{\text{i}}},\,\frac{c}{{{t_{\text{i}}}}}} \right),\,{\text{i}} = 1,\,2,\,3,\,4$$ then :

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

Practice More MCQ Question on Maths Section

If the coordinates of four concyclic points on the rectangular hyperbola $$xy = {c^2}$$ are $$\left( {c{t_{\text{i}}},\,\frac{c}{{{t_{\text{i}}}}}} \right),\,{\text{i}} = 1,\,2,\,3,\,4$$ then :

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

Practice More MCQ Question on Maths Section

Releted MCQ Question on
Geometry >> Hyperbola

Practice More Releted MCQ Question on
Hyperbola