Question

A conic section is defined by the equations x = - 1 + sect y = 2 + z tant. The coordinates of the foci are,

• A. $(-1-\sqrt{10},2)and(-1+\sqrt{10},2)$
• B. $(-1-\sqrt{8},2)and(-1+\sqrt{8},2)$
• C. $(-1,2-\sqrt{8})and(-1,2+\sqrt{8})$
• D. $(-1,2-\sqrt{10})and(-1,2+\sqrt{10})$

Answer 1

The correct option is A

$(-1-\sqrt{10},2)and(-1+\sqrt{10},2)$

The parametric form of the conic section is given as (-1+sec t,2+3 tant)=(x,y).

i.e,sect=x+1,$tant=\frac{(y-2)}{3}$

We know

$se{c}^{2}t-ta{n}^{2}t=1$

Eliminating from this form,

$(x+1{)}^2-{\left[\frac{y-2}{3}\right]}^2=1$

This is a hyperbola with origin at (-1,2)

$a^2=1$

$b^2=9$

$e^2=1+\frac{b^2}{a^2}=1+9$

=10

$\therefore Coordinatesoffocus=(\pm ae-1,0+2)$

$(\pm \sqrt{10}-1,+2)$

$\therefore fociare(-1+\sqrt{10},2)and(-1-\sqrt{10},2)$