1. Consider a branch of the hyperbola \[x^{2}-2y^{2}-2\sqrt{2}x-4\sqrt{2}y-6=0\]

with vertex at the point A.
Let B be one of the end points of its latus rectum. If C is
the focus of the hyperbola nearest to the point A, the area
of the triangle ABC is

a) \[1-\sqrt{2/3}\]

b) \[\sqrt{3/2}-1\]

c) \[1+\sqrt{2/3}\]

d) \[\sqrt{3/2}+1\]

Explanation: Equation of the branch of the hyperbola can be written as

2. The tangent at any point P( a sec \[\theta\] , b tan \[\theta\] )
of the hyperbola \[x^{2}/a^{2}-y^{2}/b^{2}=1\] makes an intercept of
length p between the point of contact and the transvers axis
of the hyperbola. p_{1}, p_{2} are the lengths of the perpendiculars
drawn from the foci on the normal at P, then

a) p is an arithmetic mean between \[P_{1}\] and \[P_{2}\]

b) p is a geometric mean between \[P_{1}\] and \[P_{2}\]

c) p is a harmonic mean between \[P_{1}\] and \[P_{2}\]

d) none of these

Explanation:

3. Let a> 0 and A(-a,0) and (a,0) be two fixed points. Let a point
P moves such that base angles
of the triangle PAB be such then \[\angle PAB=2\angle PBA \] , then P
traces

a) a straight lines

b) a circle

c) an ellipse

d) a hyperbola

Explanation: Suppose coordinates of P be (h, k) Slope of

4. Suppose base BC of a triangle is of
fixed lenth "a" and its vertex A moves such that the ratio \[\frac{\tan\left(B/2\right)}{\tan\left(C/2\right)}\] is a constant \[k\neq 1\] , then A traces

a) a circle

b) an ellipse

c) a hyperbola

d) a parabola

Explanation:

5. \[x=a\cos\theta+b\sin\theta,y=a\sin\theta-b\cos\theta , a,b>0\]

represents a rectangular hyperbola if

a) \[0<\theta<\frac{\pi}{4}\]

b) \[\frac{\pi}{4}<\theta<\frac{3\pi}{4}\]

c) \[0\leq\theta\leq\frac{\pi}{2}\]

d) \[\frac{3\pi}{2}<\theta<\pi\]

Explanation:

6. Tangent at a point P to the rectangular
hyperbola xy= 20 where it intersects the rectangular
hyperbola \[x^{2}-y^{2}=9\] is

a) \[4x+5y\pm40=0\]

b) \[4x\pm5y+40=0\]

c) \[4x\pm5y-40=0\]

d) \[4x+5y\pm20=0\]

Explanation: For the coordinates of P,

7. If x+ y =b is a tangent to the hyperbola
xy= 9, then b equals

a) \[\pm 3\]

b) \[\pm 6\]

c) \[\pm 1\]

d) \[\pm 9\]

Explanation: An equation of the tangent at (3t, 3/t) to the curve xy = 9 is

8. 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 are

a) \[2\sqrt{2x}\pm3y+8=0\]

b) \[3x\pm2\sqrt{2y}-3=0\]

c) \[2x\pm2\sqrt{2y}+7=0\]

d) \[x\pm2\sqrt{2y}+8=0\]

Explanation: Let (h, k) be the point such that the chord of contact of (h, k) with respect to the hyperbola x

^{2}– y

^{2}= 9 is x = 9. We know that chord of contact of (h, k) with respect to x

^{2}– y

^{2}= 9 is T = 0, i.e., hx – ky – 9 = 0

9. Let \[A_{i}\left(t_{i},\frac{c}{t_{i}}\right)i=1,2,3\] be three points
on the rectangular hyperbola \[xy=c^{2}\] , then the orthocenter of
the triangle \[A_{1}A_{2}A_{3}\] lies on

a) x-axis

b) y-axis

c) \[xy=c^{2}\]

d) \[x^{2}-y^{2}=c^{2}\]

Explanation:

10. Area of the triangle formed by a tangent
to the hyperbola xy= 75 and its asymptotes is

a) \[25\left(unit\right)^{2}\]

b) \[50\left(unit\right)^{2}\]

c) \[100\left(unit\right)^{2}\]

d) \[150\left(unit\right)^{2}\]

Explanation:

hyperbola is