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1. If foci of hyperbola lie on y = x and one of the asymptote is y = 2x, then equation of the hyperbola, given that it passes through (3, 4), is

(A) x2 – y2 – xy + 5 = 0 (B) 2x2 – 2y2 + 5xy + 5 = 0

(C) 2x2 + 2y2 – 5xy + 10 = 0 (D) none of these

1. CMajor axis of hyperbola bisects the asymptote Þ equation of other asymptote x = 2yequation of hyperbola (y – 2x)(x – 2y) + k = 0 it passes through (3, 4)Þ required equation 2x2 + 2y2 – 5xy + 10 = 0.

2. The locus of the centre of a variable circle touching two circles of radius r1 and r2 externally, which also touch

each other externally, is a conic. The eccentricity of the conic, if , is

(A) 1 (B)

(C) (D)

2. B

The curve is hyperbola and e = .

3. If the foci of the ellipse and the hyperbola coincide then length of latus rectum of

ellipse is

(A) (B)

(C) (D) none of these

3. BLet eccentricity of ellipse and hyperbola be e1 and e2 respectively, then b2 = 16 and 81 = 144

Þ and Þ e2 =

Now given 4e1 = Þ 4 = 3

.

Latus rectum of ellipse = .

4. A normal to the hyperbola = 1, has equal intercepts on positive x and y axes. If this normal touches

the ellipse = 1, then a2 + b2 is equal to

(A) 5 (B) 25(C) 16 (D) none of these

4. D

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The equation of the normal to the hyperbola at (2 sec q, tan q) is

2x cos q + y cot q = 5

Slope of the normal = - 2 sin q = - 1 Þ sin q =

Þ q = .

Y-intercept of the normal =

Since it touches the ellipse

Þ a2 + b2 =

5. If the variable curves y2 = 4(x2 + a) and y = bx3 + c (a and c are variable parameter and b is constant) touch each other, then their point of contact lie on a (A) straight line (B) circle(C) parabola (D) hyperbola

5. Dy2 = 4(x2 + a), y = bx3 + c

2y = 8x, = 3bx2

at the point where curve touches each other, we should have

= 3bx2 Þ xy =

So, points of contact will lie on the hyperbola (rectangular hyperbola).

6. If values of m for which the line y = mx + 2 touches the hyperbola 16x2 – 9y2 = 144 are the roots of the equation x2 – (a + b)x – 4 = 0, then value of (a + b) is equal to(A) 2 (B) 4(C) zero (D) none of these

6. C

Equation of hyperbola .

Equation of tangent y = mx +

Þ = 2 Þ m = ± 2Þ a + b = sum of roots = 0.

5. If tangents OQ and OR are drawn to variable circles having radius r and the centre lying on the rectangular hyperbola xy = 1, then locus of circumcentre of triangle OQR is (O being the origin)

(A) xy = 4 (B) xy =

(C) xy = 1 (D) none of these

5. B

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Let S be point on rectangular hyperbola

say .

Now, circumcircle of D OQR also passes through S. Therefore circumcentre is the mid point of OS.

Þ x =

Þ xy = .

6. If tangents PQ and PR are drawn from variable point P to the hyperbola,

(a > b), so that the fourth vertex S of parallelogram PQSR lies on circumcircle of triangle PQR, then locus of P is(A) x2 + y2 = b2 (B) x2 + y2 = a2

(C) x2 + y2 = (a2 - b2) (D) none of these

6. CFourth vertex of parallelogram lies on circumcircle Þ parallelogram is cyclic Þ parallelogram is a rectangle Þ tangents are perpendicular Þ locus of P is the director circle.

7. The foot of perpendicular from the focus on any tangent to a hyperbola lies on the(A) conjugate hyperbola (B) auxiliary circle(C) one of the asymptotes (D) director circle

7. B

Assuming the hyperbola as , the equation of a tangent to this hyperbola is

y – mx = . ……(1)The equation of the perpendicular line through focus is my + x = ae. ……(2)Squaring and adding (1) and (2) we get the required locus as x2 + y2 = a2.

8. The chords of contact of a point ‘P’ w.r.t. a hyperbola and its auxiliary circle are at right angle, then the point P lies on(A) conjugate hyperbola (B) one of the directrix(C) one of the asymptotes (D) none of these

8. CLet P be (h, k) be any point Þ chord of contact of P w.r.t the hyperbola is

……(1)

Þ chord of contact of P w.r.t the auxiliary circle is hx + ky = a2 – b2 ……(2)

Now,

Þ .

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9. Four points are such that the line joining any two points is perpendicular to the line joining other two points. If three points out of these lie on a rectangular hyperbola then the fourth point will lie on(A) the same hyperbola (B) conjugate hyperbola(C) one of the directrix (D) one of the asymptotes

9. AThe points are such that one of the points is the orthocentre of the triangle formed by other three points. When the vertices of a triangle lie on a rectangular hyperbola the orthocentre also lies on the same hyperbola.

10. If the chords of contact of tangents from two points and (2, 1) to the hyperbola are at right

angles then eccentricity of the hyperbola is

(A) (B)

(C) (D)

10. B

Chords of contact are

These are at right angles

So,

Now,

11. Let P be the point of intersection of the hyperbolas in the first quadrant. If tangents at P to both the curves intersect the y-axis at Q and R then circumcentre of triangle PQR lies on(A) x + y = 1 (B) x – y = 1(C) x-axis (D) y-axis

11. DWe know hyperbola and always intersects each other at right angle.So, triangle is right angle triangle.Circumcentre lies on y-axis.

12. If the tangent at the point (a sec a, b tan a) to the hyperbola = 1 meets the transverse axis at T, then

the distance of T from a focus of hyperbola is (A) a (e ± cos a) (B) a e(C) b (e + cos a) (D)

12. Aco-ordinates of point T (a cos a, 0) so distance from focus of the point T is a (e ± cos a)

13. If a rectangular hyperbola circumscribes a triangle, then it also passes through (A) incentre of the triangle (B) circumcentre of the triangle(C) orthocentre of the triangle (D) none of these

13. C

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14. A tangent to the parabola x2 = 4 ay meets the hyperbola xy = k2 in two points P and Q. Then the middle point of PQ lies on a (A) circle (B) parabola(C) ellipse (D) hyperbola

14. BLet (a, b) be the middle point chord with T = S1

middle point y = - + 2 a

which is tangent x2 = 4 a

D = 0

b2 =

y2 = , which is parabola

15. If P is a point on a hyperbola, then locus of excentre of the circle escribed opposite to ÐP for triangle PSS¢ (S, S¢ are foci), is (A) hyperbola (B) ellipse(C) parabola (D) tangents at vertex

15. DLet h, k be excentre h =

h = – a Þ x = – a (for a sec q > 0)similarly x = a for a sec q < 0Þ locus is x2 = a2.

16. In above question locus of excentre of the circle escribed opposite to ÐS¢, is (A) hyperbola (B) ellipse(C) parabola (D) circle

16. AAgain let (h, k) be excentre opposite ÐS¢

h =

Þ h = ae sec q

k = Þ locus is hyperbola

17. Four points and lying xy = 1 are concyclic if

(A) abcd = 1 (B) abcd + 1 = 0 (C) ab = cd (D) ac = bd

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17. A

Slope of segment joining a and b is

Þ

Þ

Þ a + ab2cd = b + ba2cd Þ (a - b)(abcd - 1) = 0, a ¹ b

18. The curve xy = c (c > 0) and the circle x2 + y2 = 1 touch at two points, then distance between the points of contacts is (A) 1 (B) 2(C) 2 (D) none of these

18. BThe curve xy = c and the circle x2 + y2 = 1 touches each other so

x2 + – 1 = 0 Þ x4 – x2 + c2 = 0

will have equal roots so (– 1)2 – 4c2 = 0 Þ 4c2 = 1 Þ c2 = Þ c = .

Roots of the equations are x = ± Þ y = .

Clearly distance between the points of contact = 2 units. 19. A circle cuts rectangular hyperbola xy = 1 in the points (xr, yr), r = 1, 2, 3, 4 then

(A) y1y2y3y4 = 1 (B) x1x2x3x4 = 1(C) x1x2x3x4 = y1y2y3y4 = -1 (D) y1y2y3y4 = 0

19. (A), (B)Let x2 + y2 + 2gx + 2fy + c = 0 cuts the hyperbola at the points (pi 1/pi)i = 1,2,3,4

ÞÞ p1p2p3p4 = 1

Now, pi = xi and

Þ x1x2x2x4 = y1y2y3y4 = 1

20. Eccentricity of the hyperbola whose asymptotes are given by 3x+2y-5 = 0 and 2x-3y-5 = 0 is(A) 2 (B) 3/2(C) 2 (D) None of these

20. (A)The given asymptotes are perpendicularÞ Hyperbola is rectangularÞ e =

21. If one asymptote of the hyperbola = 1 makes an angle of 30°, with the positive direction of x – axis,

then the eccentricity of the hyperbola is

(A) (B)

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(C) (D) none of these

21. (C)

According to the problem = tan30° =

Þ 3b2 = a2 Þ 3a2 (e2-1) = a2

Þ e2 = Þ e =

22. The equation of the hyperbola in a standard form whose eccentricity is 2 and the distance between foci is 16 is(A) x2-y2 = 32 (B) x2-y2 = 16(C) x2-y2 = 64 (D) None of these

22. (A)Since distance between the foci is 16Þ 2ae = 16Þ 2 = 16

Þ a =

Hyperbola is x2-y2 = 32

23. If ( - 2)x2 + y2 = 4 represents a rectangular hyperbola then equals(A) 0 (B) 2(C) 1 (D) 3

23. (C)Condition for rectangular hyperbolaD ¹ 0 ( - 2) ¹ 0 Þ ¹ 2, ¹ 0 h2 > ab Þ (- 2) < 0 0 < < 2Coefficient of x2 + coefficient of y2= 0Þ - 2 + = 0

= 1

24. If y = mx + c is a tangent to the hyperbola x2 - 3y2 = 1, then (A) c2 = m2 (B) c2 = m2 + 1(C) c2 > m2 (D) c2 < m2

24. (D)Any tangent of slope m to the hyperbola x2 - 3y2 = 1 is

y = mx ± Þ c2 = m2 -

25. The number of point(s) outside the hyperbola from where two perpendicular tangents can be

drawn to the hyperbola is/are(A) None (B) 1(C) 2 (D) Infinite

25. ADirector circle is the locus of point of intersection of perpendicular tangents drawn to a curve. I.e. x2 + y2 = a2 – b2

Director circle of given hyperbolax2 + y2 = - 11Which is not possible.

26. If e, e¢ are the eccentricities of hyperbolas

then

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(A) e = e¢ (B) e = –e¢

(C) e e¢ = 1 (D)

26. DSince b2 = a2 (e2 – 1) and a2 = b2 (e¢2 – 1)

Þ

Þ 1 =

Þ e2 + e¢2 = e2 e¢2 Þ

27. The asymptote of a hyperbola xy = hx + ky are

Let f(x) =

Then global maxima of f (x) equals ___________ global minima of f(x) equals _______

27. Axy = hx + ky( x–k) (y–h) = hkShifting origin to (k,h) the above equation reduces to XY = hk = c2 (say)When x = X + k and y = Y + hThen the equation of the asymptotes are X = 0 and Y = 0i.e. x = k, y = h

28. The angle between the asymptotes of a hyperbola is 30°. The eccentricity of the hyperbola may be(A) 3 ± 1 (B) 3 + 1(C) 6 ± 2 (D) None of these

28. C

The asymptotes are y = ± x

Þ tan30° =

Þ a2 – b2 = ± 23 ab. Also a2 + b2 = a2e2

Þ e4 – 16e2 + 16 = 0Þ e2 = 8 ± 43Þ e = 6 ± 2.

29. The number of normals to the hyperbola = 1 from an external point is

(A) 2 (B) 4(C) 6 (D) 3

29. B

30. If e and e1 are the eccentricities of the hyperbolas xy = c2 and x2 – y2 = c2, then e2 + e12 is equal to

(A) 1 (B) 4(C) 6 (D) 8

30. BBoth are rectangular hyperbolas whose eccentricities are 2 and 2

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Þ e2 + e12 = 2 + 2 = 4

31. x-intercept of the tangent at the point (x1, y1) of the curve xy = c2, c ¹ 0, is(A) 2x1 (B) 2y1

(C) c2 (D) 2c2

31. A

32. The eccentricity of the hyperbola 5x2 – 4y2 + 20 = 0 is

(A) (B)

(C) (D)

32. (A)

The hyperbola is = 1

Its transverse axis is along y–axis.

Þ e = = = .

33. Any normal to the hyperbola xy = 5 has a slope which is(A) always negative (B) always positive(C) positive or negative (D) undefined.

33. (B)

= -

Þ Slope of normal = - =

which is always positive.

34. The line lx + my + n = 0 will be a normal to the hyperbola b2x2 – a2y2 = a2b2 if

(A) = (B) =

(C) = (D) None of these

34. CNormal to hyperbola will beax cosq + by cotq = a2 + b2

If this is same as lx + my + n = 0, comparing the coefficients,

=

secq = – , tanq =

sec2q – tan2q = 1

Þ .

35. If a and b are eccentric angles of the ends of a focal chord of the ellipse , then is equal

to

(A) (B)

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(C) (D)

35. BLet the end points of focal chord be (a cosa, b sina) and (a cosb, b sinb)Then, (acosa, bsina), (acosb, bsinb) and focus (ae, 0) all lie on one line.

Þ

Þ sina cosb – e sina = sinbcosa – esinbÞ sin(a – b) = e(sina – sinb)

Þ cos = ecos

Þ = e

Þ tan tan = .

36. The chords lx + my – 1 = 0 (l, m being parameters) of the curve x 2 – 3y2 + 3xy + 3x = 0, subtending a right angle at the origin, are concurrent at the point

(A) (B)

(C) (D) None of these

36. (B)The pair of lines x2 – 3y2 + 3xy + 3x (lx + my) = 0 are perpendicular to each other.

Þ 1 – 3 + 3l = 0 Þ l =

Hence the chords are 2x – 3 + 3my = 0, which pass through the fixed point .

37. If tangent at a point on the hyperbola = 1 cuts the axis at A and B and rectangle OAPB is completed,

then locus of point P is given by (when O is the centre of the hyperbola)

(A) = 1 (B) = 1

(C) = 1 (D) = 1

37. (A) Let the tangent

A (a cosq, 0), B (0, b cotq) P (a cosq, b cotq) = (h, k)

secq = , tanq =

Locus is .

38. The focus of rectangular hyperbola (x - h) (y - k) = p2 is (A) (h - p, k - p) (B) (h - p, k + p) (C) (h + p, k - p) (D) none of these

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38. ACS = p coordinate of S is either (h + p, k + p) or (h - p, k -p)

39. Portion of asymptote of hyperbola = 1 (between centre and the tangent at vertex) in the first quadrant

is cut by the y + (x - b) = 0 ( is a parameter) then (A) Î R (B) Î (-¥, 0) (C) Î(0, ¥) (D) none of these

39. CFrom diagram we can see that slope of line is always -ve - Î (-¥, 0) Î(0, ¥)

40. If xy = m2 - 4 be a rectangular hyperbola whose branches lies only in the 2nd and 4th quadrant then (A) |m| < 2 (B) |m| > 2 (C) |m| £ 2 (D) not possible

40. AFor the given situation xy < 0 Þ m2 - 4 < 0 Þ |m| < 2 .

41. Number of points on hyperbola from where mutually perpendicular tangents can be drawn to circle

x2 + y2 = a2 is (A) 2 (B) 3 (C) infinite (D) 4

41. DRequired points will lies on the intersection hyperbola with director circle.

42. The product of perpendiculars drawn from any point on upon its asymptote is

(A) (B)

(C) (D) none of these

42. CPoint is (a secq, b tanq)asymptotes are bx - ay-=0 and bx + ay = 0

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Now

= .

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