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Mathematics · Class 11 Science

Ch 7Conic Sections — Class 11 Mathematics, concept-first.

The Greek mathematicians Archimedes and Apollonius were among the first to study, in real depth, the family of curves now called conic sections — so named because every one of them is exactly what you get by intersecting a plane with a right circular cone.

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Standard Equation of a Parabola

Starting from the focus-directrix property with (so exactly), and choosing the midpoint of the perpendicular from focus to directrix as the origin, the algebra collapses beautifully to the standard equation of a parabola…

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Introduction

The Greek mathematicians Archimedes and Apollonius were among the first to study, in real depth, the family of curves now called conic sections — so named because every one of them is exactly what you…

7.1.1

Double Cone

A double cone (or double-napped cone) is the surface you get by spinning one straight line around another.

7.1.2

Conic Sections (the cross-sections of a double cone)

Think of the double cone as something physical you could actually cut — a cone-shaped vegetable like a carrot, or two ice-cream cones joined tip to tip.

7.1.3

Definition of a Conic Section and Eccentricity

Rather than always thinking about slicing a 3-D cone, it is far more useful for calculation to have a purely 2-D, plane-geometry definition of a conic — one written entirely in terms of distances, so…

7.1.4

Some Useful Terms of Conic Sections

Before deriving equations, it helps to fix the vocabulary that is common to every conic section — parabola, ellipse and hyperbola alike.

7.1.5

Standard Equation of a Parabola

Definition. A parabola is the locus of a point in the plane that is equidistant from a fixed point (the focus) and a fixed line (the directrix) — this is exactly the focus–directrix definition of sect…

7.1.6

Tracing of the Parabola $y^2=4ax$

Once we know the equation is (with ), we can work out the shape of the curve purely by studying the algebra, without plotting many individual points.

7.1.7

Focal Distance and Latus Rectum of a Parabola

Two of the most useful numerical facts about a parabola — its focal distance formula and its latus-rectum length — follow quickly from the standard equation.

7.1.8

Other Standard Forms of a Parabola

The derivation in section 7.1.5 assumed the parabola opens to the right, with focus on the positive -axis.

7.1.9

Parametric Form of a Parabola

Parameter, defined. If the coordinates of a moving point on a curve are expressed as functions of a single third variable, that variable is called the parameter for the curve.

7.1.10

General (Shifted) Form of a Parabola

Shifting the vertex. All the standard forms so far have their vertex fixed at the origin. If instead the vertex is shifted to a general point , while the axis of symmetry stays parallel to a coordinat…

7.1.11

Tangent to a Parabola

What is a tangent, geometrically? Picture a secant line cutting the parabola at two points, and a second point . Now let slide along the curve, getting closer and closer to .

7.1.12

Condition of Tangency for a Parabola

The question: for which values of and does the general line touch (rather than cross, or miss entirely) the parabola ? And if it does touch, at what point?

7.1.13

Tangents from a Point to a Parabola

How many tangents from an external point? From a general point in the plane (not necessarily on the parabola), consider all lines through with the tangent form . Forcing this line through :

7.2

Ellipse — Introduction

Definition. An ellipse is the locus of a point in a plane that moves so that its distance from a fixed point (the focus ) bears a constant ratio , with , to its distance from a fixed line (the directr…

+Exercise 7.2i33 questions
  1. Q29Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrics (iv) length of the latus rectum (v) d…Free
  2. Q30Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrics (iv) length of the latus rectum (v) d…Free
  3. Q31Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrics (iv) length of the latus rectum (v) d…Free
  4. Q32Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrics (iv) length of the latus rectum (v) d…Preview
  5. Q33Find the equation of the ellipse in standard form if eccentricity = 3/8 and distance between its foci = 6.Preview
  6. Q34Find the equation of the ellipse in standard form if the length of major axis 10 and the distance between foci is 8.Preview
  7. Q35Find the equation of the ellipse in standard form if distance between directrix is 18 and eccentricity is 1/3.Preview
  8. Q36Find the equation of the ellipse in standard form if minor axis is 16 and eccentricity is 1/3.Preview
  9. Q37Find the equation of the ellipse in standard form if the distance between foci is 6 and the distance between directrix is 50/3.Preview
  10. Q38Find the equation of the ellipse in standard form if the latus rectum has length 6 and foci are $(\pm 2, 0)$.Preview
  11. Q39Find the equation of the ellipse in standard form if passing through the points $(-3, 1)$ and $(2, -2)$.Preview
  12. Q40Find the equation of the ellipse in standard form if the dist. between its directrix is 10 and which passes through $(-\sqrt5, 2)$.Preview
  13. Q41Find the equation of the ellipse in standard form if eccentricity is 2/3 and passes through $(2, -5/3)$.Preview
  14. Q42Find the eccentricity of an ellipse, if the length of its latus rectum is one third of its minor axis.Preview
  15. Q43Find the eccentricity of an ellipse if the distance between its directrix is three times the distance between its foci.Preview
  16. Q44Show that the product of the lengths of the perpendicular segments drawn from the foci to any tangent line to the ellipse $x^2 /25 + y^2/16…Preview
  17. Q45A tangent having slope -1/2 to the ellipse $3x^2 + 4y^2 = 12$ intersects the X and Y axes in the points A and B respectively. If O is the or…Preview
  18. Q46Show that the line $x - y = 5$ is a tangent to the ellipse $9x^2 + 16y^2 = 144$. Find the point of contact.Preview
  19. Q48Determine whether the line $x + 3y\sqrt2 = 9$ is a tangent to the ellipse $x^2/9 + y^2/4 = 1$. If so, find the co-ordinates of the point of…Preview
  20. Q49Find k, if the line $3x + 4y + k = 0$ touches $9x^2 + 16y^2 = 144$.Preview
  21. Q50Find the equation of the tangent to the ellipse $x^2/5 + y^2/4 = 1$ passing through the point $(2, -2)$.Preview
  22. Q51Find the equation of the tangent to the ellipse $4x^2 + 7y^2 = 28$ from the point $(3, -2)$.Preview
  23. Q52Find the equation of the tangent to the ellipse $2x^2 + y^2 = 6$ from the point $(2, 1)$.Preview
  24. Q53Find the equation of the tangent to the ellipse $x^2 + 4y^2 = 9$ which are parallel to the line $2x + 3y - 5 = 0$.Preview
  25. Q54Find the equation of the tangent to the ellipse $x^2/25 + y^2/4 = 1$ which are parallel to the line $x + y + 1 = 0$.Preview
  26. Q55Find the equation of the tangent to the ellipse $5x^2 + 9y^2 = 45$ which are perpendicular to the line $3x + 2y = 0$.Preview
  27. Q56Find the equation of the tangent to the ellipse $x^2 + 4y^2 = 20$, perpendicular to the line $4x+3y = 7$.Preview
  28. Q57Find the equation of the locus of a point the tangents from which to the ellipse $3x^2 + 5y^2 = 15$ are at right angles.Preview
  29. Q59Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an…Preview
  30. Q60P and Q are two points on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} = 1$ with eccentric angles $\theta_1$ and $\theta_2$. Find the equa…Preview
  31. Q61The eccentric angles of two points P and Q the ellipse $4x^2 + y^2 = 4$ differ by $2\pi/3$. Show that the locus of the point of intersection…Preview
  32. Q62Find the equations of the tangents to the ellipse $x^2/16 + y^2/9 = 1$, making equal intercepts on co-ordinate axes.Preview
  33. Q63A tangent having slope – 1/2 to the ellipse $3x^2 + 4y^2 = 12$ intersects the X and Y axes in the points A and B respectively. If O is the o…Preview
7.2.1

Standard Equation of an Ellipse

Deriving the standard equation (with ).

7.2.2

Focal Properties and Latus Rectum of an Ellipse

This section collects the key derived facts about the standard ellipse, each following from the defining relations and the two-focus, two-directrix picture of section 7.2.1.

7.2.3

Special Cases of an Ellipse

The standard ellipse (with , ) has a natural limiting/special case worth noting.

7.2.4

Tangent to an Ellipse

Definition (as for the parabola). A tangent to an ellipse is a straight line that intersects the curve in two coincident points — equivalently, the limiting position of a secant as its two points of i…

7.2.5

Condition for Tangency of an Ellipse

The question: for which does touch the ellipse , and where?

7.2.6

Tangents from a Point to the Ellipse

From a point in the plane, using the slope-form tangent from section 7.2.5, and forcing it through :

7.2.7

Director Circle: Locus of Perpendicular Tangents (Ellipse)

Setting up the perpendicularity condition. From section 7.2.6, the two tangent slopes from satisfy . If the two tangents are mutually perpendicular, then :

7.2.8

Auxiliary Circle and Director Circle of an Ellipse

Two circles are naturally associated with every ellipse, and it is worth keeping their definitions and equations clearly distinct:

7.3

Hyperbola — Introduction

Definition. A hyperbola is the locus of a point in a plane that moves so that its distance from a fixed point (the focus ) bears a constant ratio , with , to its distance from a fixed line (the direct…

+Exercise 7.3i31 questions
  1. Q64Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Free
  2. Q65Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Free
  3. Q66Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Free
  4. Q67Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Preview
  5. Q68Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Preview
  6. Q69Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Preview
  7. Q70Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Preview
  8. Q71Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Preview
  9. Q72Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Preview
  10. Q73Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the l…Preview
  11. Q74Find the equation of the hyperbola with centre at the origin, length of conjugate axis 10 and one of the foci $(-7,0)$.Preview
  12. Q75Find the eccentricity of the hyperbola, which is conjugate to the hyperbola $x^2 - 3y^2 = 3$.Preview
  13. Q76If e and e' are the eccentricities of a hyperbola and its conjugate hyperbola respectively, prove that $\dfrac{1}{e^2}+\dfrac{1}{e'^2}=1$.Preview
  14. Q77Find the equation of the hyperbola referred to its principal axes whose distance between foci is 10 and eccentricity $\dfrac{5}{2}$.Preview
  15. Q78Find the equation of the hyperbola referred to its principal axes whose distance between foci is 10 and length of conjugate axis 6.Preview
  16. Q79Find the equation of the hyperbola referred to its principal axes whose distance between directrices is $\dfrac{8}{3}$ and eccentricity is $…Preview
  17. Q80Find the equation of the hyperbola referred to its principal axes whose length of conjugate axis = 12 and passing through $(11, -2)$.Preview
  18. Q81Find the equation of the hyperbola referred to its principal axes which passes through the points $(6,9)$ and $(3,0)$.Preview
  19. Q82Find the equation of the hyperbola referred to its principal axes whose vertices are $(\pm7,0)$ and end points of conjugate axis are $(0, \p…Preview
  20. Q83Find the equation of the hyperbola referred to its principal axes whose foci are at $(\pm2,0)$ and eccentricity $\dfrac{3}{2}$.Preview
  21. Q84Find the equation of the hyperbola referred to its principal axes whose length of transverse and conjugate axis are 6 and 9 respectively.Preview
  22. Q85Find the equation of the hyperbola referred to its principal axes whose length of transverse axis is 8 and distance between foci is 10.Preview
  23. Q86Find the equation of the tangent to the hyperbola $3x^2 - y^2 = 4$ at the point $(2, 2\sqrt2)$.Preview
  24. Q87Find the equation of the tangent to the hyperbola $3x^2 - 4y^2 = 12$ at the point $(4,3)$.Preview
  25. Q88Find the equation of the tangent to the hyperbola $\dfrac{x^2}{144} - \dfrac{y^2}{25} = 1$ at the point whose eccentric angle is $\dfrac{\pi…Preview
  26. Q89Find the equation of the tangent to the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$ at the point in the first quadrant whose ordinate i…Preview
  27. Q90Find the equation of the tangent to the hyperbola $9x^2 - 16y^2 = 144$ at the point L of latus rectum in the first quadrant.Preview
  28. Q91Show that the line $3x - 4y + 10 = 0$ is tangent to the hyperbola $x^2 - 4y^2 = 20$. Also find the point of contact.Preview
  29. Q92If the $3x - 4y = k$ touches the hyperbola $\dfrac{x^2}{5} - \dfrac{4y^2}{5} = 1$ then find the value of k.Preview
  30. Q93Find the equations of the tangents to the hyperbola $\dfrac{x^2}{25} - \dfrac{y^2}{9} = 1$ making equal intercepts on the co-ordinate axes.Preview
  31. Q94Find the equations of the tangents to the hyperbola $5x^2 - 4y^2 = 20$ which are parallel to the line $3x + 2y + 12 = 0$.Preview
7.3.1

Standard Equation of a Hyperbola

Deriving the standard equation .

7.3.2

Some Useful Terms of the Hyperbola

A few structural facts about the hyperbola that distinguish it from the ellipse:

7.3.3

Focal Properties, Latus Rectum and Parametric Form of a Hyperbola

1. Distance between directrices. Exactly as for the ellipse: .

7.3.4

Tangent to a Hyperbola

Definition (as before). A tangent to a hyperbola is a straight line intersecting the curve in two coincident points.

7.3.5

Condition for Tangency of a Hyperbola

The question: for which does touch ?

7.3.6

Tangents from a Point to the Hyperbola

From an external point , force the slope-form tangent through :

7.3.7

Director Circle: Locus of Perpendicular Tangents (Hyperbola)

Setting the two tangent slopes' product (perpendicular tangents), using from section 7.3.6:

7.3.8

Auxiliary Circle and Director Circle of a Hyperbola

As with the ellipse, two circles are naturally associated with the hyperbola:

7.3.9

Asymptotes of a Hyperbola

What is an asymptote? Consider the two lines through the origin, (equivalently ). Now imagine a point moving along the hyperbola, further and further from the centre.

7.4

Summary: Conics Compared

This closing section gathers the chapter's results into two comparison tables and a short recap, so that the parallels and the (few but important) sign differences between the three conics are visible…

More questions

87 Q
+Show 28 questions28 questions
  1. Q1Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola $…Free
  2. Q2Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola $…Free
  3. Q3Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola $…Free
  4. Q4Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola $…Preview
  5. Q5Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola $…Preview
  6. Q6Find the equation of the parabola with vertex at the origin, axis along Y-axis and passing through the point $(-10,-5)$.Preview
  7. Q7Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point $(3,4)$.Preview
  8. Q8Find the equation of the parabola whose vertex is O(0,0) and focus at $(-7,0)$.Preview
  9. Q9Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point $(1,-6)$.Preview
  10. Q10Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point $(2,3)$.Preview
  11. Q11For the parabola $3y^2 = 16x$, find the parameter of the point $(3,-4)$.Preview
  12. Q12For the parabola $3y^2 = 16x$, find the parameter of the point $(27,-12)$.Preview
  13. Q13Find the focal distance of a point on the parabola $y^2 = 16x$ whose ordinate is 2 times the abscissa.Preview
  14. Q14Find coordinate of the point on the parabola $y^2=12x$ whose parameter is $1/3$. Also find focal distance.Preview
  15. Q15Find coordinate of the point on the parabola $2y^2=7x$ whose parameter is $-2$. Also find focal distance.Preview
  16. Q16For the parabola $y^2 = 4x$, find the coordinate of the point whose focal distance is 17.Preview
  17. Q17Find length of latus rectum of the parabola $y^2 = 4ax$ passing through the point $(2,-6)$.Preview
  18. Q18Find the area of the triangle formed by the line joining the vertex of the parabola $x^2 = 12y$ to the end points of latus rectum.Preview
  19. Q19If a parabolic reflector is 20cm in diameter and 5 cm deep, find its focus.Preview
  20. Q20Find coordinate of focus, vertex and equation of directrix and the axis of the parabola $y = x^2 - 2x + 3$.Preview
  21. Q21Find the equation of tangent to the parabola $y^2 = 12x$ from the point $(2,5)$.Preview
  22. Q22Find the equation of tangent to the parabola $y^2 = 36x$ from the point $(2,9)$.Preview
  23. Q23If the tangent drawn from the point $(-6,9)$ to the parabola $y^2 = kx$ are perpendicular to each other, find k.Preview
  24. Q24Two tangents to the parabola $y^2 = 8x$ meet the tangents at the vertex in the point P and Q. If PQ = 4, prove that the equation of the locu…Preview
  25. Q25Find the equation of common tangent to the parabola $y^2 = 4x$ and $x^2 = 32y$.Preview
  26. Q26Find the equation of the locus of a point, the tangents from which to the parabola $y^2 = 18x$ are such that some of their slopes is -3.Preview
  27. Q27The tower of a bridge, hung in the form of a parabola have their tops 30 meters above the road way and are 200 meters apart. If the cable is…Preview
  28. Q28A circle whose centre is $(4,-1)$ passes through the focus of the parabola $x^2 + 16y = 0$. Show that the circle touches the directrix of th…Preview
+Show 20 questions20 questions
  1. Q95The line $y = mx + 1$ is tangent to the parabola $y^2 = 4x$ if m is A) 1 B) 2 C) 3 D) 4Free
  2. Q96The length of latus rectum of the parabola $x^2 - 4x - 8y + 12 = 0$ is A) 4 B) 6 C) 8 D) 10Free
  3. Q97If the focus of the parabola is $(0,-3)$ its directrix is $y = 3$ then its equation is A) $x^2=-12y$ B) $x^2=12y$ C) $y^2=12x$ D) $y^2=-12x$Free
  4. Q98The coordinates of a point on the parabola $y^2 = 8x$ whose focal distance is 4 are A) (1/2, ±2) B) (1, ±2√2) C) (2, ±4) D) none of thesePreview
  5. Q99The end points of latus rectum of the parabola $y^2 = 24x$ are A) (6, ± 12) B) (12, ±6) C) (6, ± 6) D) none of thesePreview
  6. Q100Equation of the parabola with vertex at the origin and directrix $x + 8 = 0$ is A) $y^2 = 8x$ B) $y^2 = 32x$ C) $y^2 = 16x$ D) $x^2 = 32y$Preview
  7. Q101The area of the triangle formed by the line joining the vertex of the parabola $x^2 = 12y$ to the end points of its latus rectum is A) 22 sq…Preview
  8. Q102If $P\left(\dfrac{\pi}{4}\right)$ is any point on the ellipse $9x^2 + 25y^2 = 225$. S and S' are its foci then $SP \cdot S'P =$ A) 13 B) 14…Preview
  9. Q103The equation of the parabola having $(2, 4)$ and $(2, -4)$ as end points of its latus rectum is A) $y^2 = 4x$ B) $y^2 = 8x$ C) $y^2 = -16x$…Preview
  10. Q104If the parabola $y^2 = 4ax$ passes through $(3, 2)$ then the length of its latus rectum is A) 2/3 B) 4/3 C) 1/3 D) 4Preview
  11. Q105The eccentricity of rectangular hyperbola is A) 1/2 B) $1/\sqrt2$ C) $\sqrt2$ D) $1/\sqrt3$Preview
  12. Q106The equation of the ellipse having foci $(\pm 4, 0)$ and eccentricity $1/3$ is, A) $9x^2 + 16y^2 = 144$ B) $144x^2 + 9y^2 = 1296$ C) $128x^2…Preview
  13. Q107The equation of the ellipse having eccentricity $\dfrac{\sqrt3}{2}$ and passing through $\left(-\sqrt8, \sqrt3\right)$ is A) $4x^2 + y^2 = 4…Preview
  14. Q108If the line $4x - 3y + k = 0$ touches the ellipse $5x^2 + 9y^2 = 45$ then the value of k is A) ± 21 B) $\pm3\sqrt{21}$ C) $\pm3$ D) $\pm3\sq…Preview
  15. Q109The equation of the ellipse is $16x^2 + 25y^2 = 400$. The equations of the tangents making an angle of 180° with the major axis are A) x = 4…Preview
  16. Q110The equation of the tangent to the ellipse $4x^2 + 9y^2 = 36$ which is perpendicular to the $3x + 4y = 17$ is, A) $y = 4x + 6$ B) $3y + 4x =…Preview
  17. Q111Eccentricity of the hyperbola $16x^2 - 3y^2 - 32x - 12y - 44 = 0$ is A) $\sqrt{17}/3$ B) $\sqrt{19}/3$ C) $\sqrt{19}/3$ D) $\sqrt{17}/3$Preview
  18. Q112Centre of the ellipse $9x^2 + 5y^2 - 36x - 50y - 164 = 0$ is at A) (2, 5) B) (1, −2) C) (−2, 1) D) (0, 0)Preview
  19. Q113If the line $2x - y = 4$ touches the hyperbola $4x^2 - 3y^2 = 24$, the point of contact is A) (1, 2) B) (2, 3) C) (3, 2) D)(−2, −3)Preview
  20. Q114The foci of hyperbola $4x^2 - 9y^2 - 36 = 0$ are A) $(\pm\sqrt{13}, 0)$ B) $(\pm\sqrt{11}, 0)$ C) $(\pm\sqrt{12}, 0)$ D) $(0, \pm\sqrt{12})$Preview
+Show 39 questions39 questions
  1. Q115For each of the following parabolas, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum: $2y^2…Free
  2. Q116For each of the following parabolas, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum: $5x^2…Free
  3. Q117Find the Cartesian co-ordinates of the points on the parabola $y^2 = 12x$ whose parameter is 2.Free
  4. Q118Find the Cartesian co-ordinates of the points on the parabola $y^2 = 12x$ whose parameter is $-3$.Preview
  5. Q119Find the co-ordinates of a point of the parabola $y^2 = 8x$ having focal distance 10.Preview
  6. Q120Find the equation of the tangent to the parabola $y^2 = 9x$ at the point $(4, -6)$ on it.Preview
  7. Q121Find the equation of the tangent to the parabola $y^2 = 8x$ at $t = 1$ on it.Preview
  8. Q122Find the equations of the tangents to the parabola $y^2 = 9x$ through the point $(4,10)$.Preview
  9. Q123Show that the two tangents drawn to the parabola $y^2 = 24x$ from the point $(-6,9)$ are at the right angle.Preview
  10. Q124Find the equation of the tangent to the parabola $y^2 = 8x$ which is parallel to the line $2x + 2y + 5 = 0$. Find its point of contact.Preview
  11. Q125A line touches the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$. Show that its equation is $y = \pm(x+2)$.Preview
  12. Q126Two tangents to the parabola $y^2 = 8x$ meet the tangent at the vertex in P and Q. If PQ = 4, prove that the locus of the point of intersect…Preview
  13. Q127The slopes of the tangents drawn from P to the parabola $y^2 = 4ax$ are $m_1$ and $m_2$. Using the quadratic satisfied by the slopes, state…Preview
  14. Q128The tangent at point P on the parabola $y^2 = 4ax$ meets the y-axis in Q. If S is the focus, show that SP subtends a right angle at Q.Preview
  15. Q129Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrices (iv) length of the latus rectum (v)…Preview
  16. Q130Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrices (iv) length of the latus rectum (v)…Preview
  17. Q131Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrices (iv) length of the latus rectum (v)…Preview
  18. Q132Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrices (iv) length of the latus rectum (v)…Preview
  19. Q133Find the equation of the ellipse in standard form if eccentricity = 3/8 and distance between its foci = 6.Preview
  20. Q134Find the equation of the ellipse in standard form if the length of major axis 10 and the distance between foci is 8.Preview
  21. Q135Find the equation of the ellipse in standard form if passing through the points $(-3, 1)$ and $(2, -2)$.Preview
  22. Q136Find the eccentricity of an ellipse if the distance between its directrices is three times the distance between its foci.Preview
  23. Q137For the hyperbola $x^2/100 - y^2/25 = 1$, prove that $SA \cdot S'A = 25$, where S and S' are the foci and A is the vertex.Preview
  24. Q138Find the equation of the tangent to the ellipse $x^2/5 + y^2/4 = 1$ passing through the point $(2,-2)$.Preview
  25. Q139Find the equation of the tangent to the ellipse $x^2 + 4y^2 = 100$ at $(8,3)$.Preview
  26. Q140Show that the line $8y + x = 17$ touches the ellipse $x^2 + 4y^2 = 17$. Find the point of contact.Preview
  27. Q140Show that the line $8y + x = 17$ touches the ellipse $x^2 + 4y^2 = 17$. Find the point of contact.Preview
  28. Q141Tangents are drawn through a point P to the ellipse $4x^2 + 5y^2 = 20$ having inclinations $\theta_1$ and $\theta_2$ such that $\tan\theta_1…Preview
  29. Q141Tangents are drawn through a point P to the ellipse $4x^2 + 5y^2 = 20$ having inclinations $\theta_1$ and $\theta_2$ such that $\tan\theta_1…Preview
  30. Q142Show that the product of the lengths of its perpendicular segments drawn from the foci to any tangent line to the ellipse $x^2/25 + y^2/16 =…Preview
  31. Q143Find the equation of the hyperbola in the standard form if Length of conjugate axis is 5 and distance between foci is 13.Preview
  32. Q144Find the equation of the hyperbola in the standard form if eccentricity is 3/2 and distance between foci is 12.Preview
  33. Q145Find the equation of the hyperbola in the standard form if length of the conjugate axis is 3 and distance between the foci is 5.Preview
  34. Q146Find the equation of the tangent to the hyperbola $7x^2 - 3y^2 = 51$ at $(-3, -2)$.Preview
  35. Q147Find the equation of the tangent to the hyperbola $x = 3\sec\theta,\ y = 5\tan\theta$ at $\theta = \pi/3$.Preview
  36. Q148Find the equation of the tangent to the hyperbola $x^2/25 - y^2/16 = 1$ at $P(30°)$.Preview
  37. Q149Show that the line $2x - y = 4$ touches the hyperbola $4x^2 - 3y^2 = 24$. Find the point of contact.Preview
  38. Q150Find the equations of the tangents to the hyperbola $3x^2 - y^2 = 48$ which are perpendicular to the line $x + 2y - 7 = 0$.Preview
  39. Q151Two tangents to the hyperbola make angles $\theta_1, \theta_2$, with the transverse axis. Find the locus of their point of intersection if $…Preview