Hyperbola Questions and Answers Part-2

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

Answer: b
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. p1, p2 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

Answer: c

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

Answer: d
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

Answer: c

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

Answer: b

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

Answer: a
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) \[\pm 9\]

Answer: b
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) \[x\pm2\sqrt{2y}+8=0\]

Answer: b
Explanation: Let (h, k) be the point such that the chord of contact of (h, k) with respect to the hyperbola x2 – y2 = 9 is x = 9. We know that chord of contact of (h, k) with respect to x2 – y2 = 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) \[x^{2}-y^{2}=c^{2}\]

Answer: c

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

Answer: d
hyperbola is
q20a q20b