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Mathematics · Class 12 Science

Ch 8Conic Sections — Class 12 Mathematics, concept-first.

In the previous chapter, you mastered the straight line — its equations, slopes, and intercepts. But the world of curves is far richer. The circle, the ellipse, the parabola, and the hyperbola are not just abstract shapes; they are the paths of planets, the curves that focus light in telescopes and flashlights, and the…

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Circle Equation Standard Form

Imagine you're standing at a point on a flat field. You tie a rope to a stake at that point, walk out the full length of the rope, and start walking in a circle, keeping the rope taut.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

10.1

Introduction

In the previous chapter, you mastered the straight line — its equations, slopes, and intercepts. But the world of curves is far richer.

10.2

Sections of a Cone

Before we cut anything, we need a clear picture of the object being cut. Take a fixed vertical line . Now take another line that intersects at a fixed point , and is tilted away from by a fixed angle…

10.2.1

Circle, Ellipse, Parabola and Hyperbola

The shape you get when a plane cuts through a double cone depends entirely on the angle at which the plane meets the cone's axis.

10.2.2

Degenerated Conic Sections

Up to this point, we have considered planes that cut through a double-napped cone at various angles, producing the familiar curves: circles, ellipses, parabolas, and hyperbolas.

10.3

Circle

19 Q

A circle is the set of all points in a plane that are at a fixed distance from a fixed point in that plane.

+Worked Examplesi4 questions
  1. Example 1Find an equation of the circle with centre at $(0, 0)$ and radius $r$.Free
  2. Example 2Find the equation of the circle with centre $(-3, 2)$ and radius $4$.Free
  3. Example 3Find the centre and the radius of the circle $x^2 + y^2 + 8x + 10y - 8 = 0$.Preview
  4. Example 4Find the equation of the circle which passes through the points $(2, -2)$ and $(3, 4)$ and whose centre lies on the line $x + y = 2$.Preview
+Exercise 10.1i15 questions
  1. Q1Find the equation of the circle with centre $(0, 2)$ and radius $2$.Free
  2. Q2Find the equation of the circle with centre $(-2, 3)$ and radius $4$.Free
  3. Q3Find the equation of the circle with centre $\left(\frac{1}{2}, \frac{1}{4}\right)$ and radius $\frac{1}{12}$.Free
  4. Q4Find the equation of the circle with centre $(1, 1)$ and radius $\sqrt{2}$.Preview
  5. Q5Find the equation of the circle with centre $(-a, -b)$ and radius $\sqrt{a^2 - b^2}$.Preview
  6. Q6Find the centre and radius of the circle $(x + 5)^2 + (y - 3)^2 = 36$.Preview
  7. Q7Find the centre and radius of the circle $x^2 + y^2 - 4x - 8y - 45 = 0$.Preview
  8. Q8Find the centre and radius of the circle $x^2 + y^2 - 8x + 10y - 12 = 0$.Preview
  9. Q9Find the centre and radius of the circle $2x^2 + 2y^2 - x = 0$.Preview
  10. Q10Find the equation of the circle passing through the points $(4, 1)$ and $(6, 5)$ and whose centre is on the line $4x + y = 16$.Preview
  11. Q11Find the equation of the circle passing through the points $(2, 3)$ and $(-1, 1)$ and whose centre is on the line $x - 3y - 11 = 0$.Preview
  12. Q12Find the equation of the circle with radius $5$ whose centre lies on $x$-axis and passes through the point $(2, 3)$.Preview
  13. Q13Find the equation of the circle passing through $(0, 0)$ and making intercepts $a$ and $b$ on the coordinate axes.Preview
  14. Q14Find the equation of a circle with centre $(2, 2)$ and passes through the point $(4, 5)$.Preview
  15. Q15Does the point $(-2.5, 3.5)$ lie inside, outside or on the circle $x^2 + y^2 = 25$?Preview
10.4

Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane.

10.4.1

Standard Equations of Parabola

The simplest and most useful form of a parabola's equation occurs when its vertex is placed at the origin and its axis of symmetry lies along either the x-axis or the y-axis.

10.4.2

Latus Rectum

16 Q

The latus rectum is a special line segment that helps us measure the "width" of a parabola at its focus.

+Worked Examplesi4 questions
  1. Example 5Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola $y^2 = 8x$.Free
  2. Example 6Find the equation of the parabola with focus $(2, 0)$ and directrix $x = -2$.Free
  3. Example 7Find the equation of the parabola with vertex at $(0, 0)$ and focus at $(0, 2)$.Preview
  4. Example 8Find the equation of the parabola which is symmetric about the $y$-axis, and passes through the point $(2, -3)$.Preview
+Exercise 10.2i12 questions
  1. Q1Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $y…Free
  2. Q2Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $x…Free
  3. Q3Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $y…Free
  4. Q4Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $x…Preview
  5. Q5Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $y…Preview
  6. Q6Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $x…Preview
  7. Q7Find the equation of the parabola that satisfies the given conditions: Focus $(6, 0)$; directrix $x = -6$.Preview
  8. Q8Find the equation of the parabola that satisfies the given conditions: Focus $(0, -3)$; directrix $y = 3$.Preview
  9. Q9Find the equation of the parabola that satisfies the given conditions: Vertex $(0, 0)$; focus $(3, 0)$.Preview
  10. Q10Find the equation of the parabola that satisfies the given conditions: Vertex $(0, 0)$; focus $(-2, 0)$.Preview
  11. Q11Find the equation of the parabola that satisfies the given conditions: Vertex $(0, 0)$ passing through $(2, 3)$ and axis is along $x$-axis.Preview
  12. Q12Find the equation of the parabola that satisfies the given conditions: Vertex $(0, 0)$, passing through $(5, 2)$ and symmetric with respect…Preview
10.5

Ellipse

An ellipse is the set of all points in a plane for which the sum of the distances to two fixed points remains constant. Those two fixed points are called the foci (singular: focus) of the ellipse.

10.5.1

Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus From the Centre of the Ellipse

Every ellipse is defined by three key numbers: the semi-major axis , the semi-minor axis , and the distance from the centre to each focus.

10.5.2

Eccentricity

An ellipse is defined by its two fixed foci and the constant sum of distances property. But how stretched is it? That is what eccentricity measures.

10.5.3

Standard Equations of an Ellipse

The equation of an ellipse takes its simplest form when the centre is at the origin and the foci lie on one of the coordinate axes.

10.5.4

Latus Rectum

25 Q

The latus rectum is a line segment that passes through a focus of the ellipse and is perpendicular to the major axis. Its endpoints lie on the ellipse itself.

+Worked Examplesi5 questions
  1. Example 9Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the latus rectum of the ellip…Free
  2. Example 10Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse $9x^2 + 4y^2 = 36$.Free
  3. Example 11Find the equation of the ellipse whose vertices are $(\pm 13, 0)$ and foci are $(\pm 5, 0)$.Preview
  4. Example 12Find the equation of the ellipse, whose length of the major axis is $20$ and foci are $(0, \pm 5)$.Preview
  5. Example 13Find the equation of the ellipse, with major axis along the $x$-axis and passing through the points $(4, 3)$ and $(-1, 4)$.Preview
+Exercise 10.3i20 questions
  1. Q1Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Free
  2. Q2Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Free
  3. Q3Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Free
  4. Q4Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Preview
  5. Q5Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Preview
  6. Q6Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Preview
  7. Q7Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Preview
  8. Q8Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Preview
  9. Q9Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectu…Preview
  10. Q10Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 5, 0)$, foci $(\pm 4, 0)$.Preview
  11. Q11Find the equation for the ellipse that satisfies the given conditions: Vertices $(0, \pm 13)$, foci $(0, \pm 5)$.Preview
  12. Q12Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 6, 0)$, foci $(\pm 4, 0)$.Preview
  13. Q13Find the equation for the ellipse that satisfies the given conditions: Ends of major axis $(\pm 3, 0)$, ends of minor axis $(0, \pm 2)$.Preview
  14. Q14Find the equation for the ellipse that satisfies the given conditions: Ends of major axis $(0, \pm \sqrt{5})$, ends of minor axis $(\pm 1, 0…Preview
  15. Q15Find the equation for the ellipse that satisfies the given conditions: Length of major axis $26$, foci $(\pm 5, 0)$.Preview
  16. Q16Find the equation for the ellipse that satisfies the given conditions: Length of minor axis $16$, foci $(0, \pm 6)$.Preview
  17. Q17Find the equation for the ellipse that satisfies the given conditions: Foci $(\pm 3, 0)$, $a = 4$.Preview
  18. Q18Find the equation for the ellipse that satisfies the given conditions: $b = 3$, $c = 4$, centre at the origin; foci on the $x$-axis.Preview
  19. Q19Find the equation for the ellipse that satisfies the given conditions: Centre at $(0, 0)$, major axis on the $y$-axis and passes through the…Preview
  20. Q20Find the equation for the ellipse that satisfies the given conditions: Major axis on the $x$-axis and passes through the points $(4, 3)$ and…Preview
10.6

Hyperbola

A hyperbola is the set of all points in a plane for which the difference of the distances from two fixed points is constant.

10.6.1

Eccentricity

The eccentricity of a hyperbola is defined in the same spirit as it was for an ellipse — it measures how “stretched” the curve is.

10.6.2

Standard Equation of Hyperbola

The simplest equation for a hyperbola arises when we place its centre at the origin and align its foci along one of the coordinate axes.

10.6.3

Latus Rectum

18 Q

The latus rectum is a line segment that runs perpendicular to the transverse axis, passing through one of the foci, with its endpoints lying on the hyperbola.

+Worked Examplesi3 questions
  1. Example 14Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus rectum of the hyperbolas: (i) $\frac{x^2}{9} -…Free
  2. Example 15Find the equation of the hyperbola with foci $(0, \pm 3)$ and vertices $\left(0, \pm \frac{\sqrt{11}}{2}\right)$.Preview
  3. Example 16Find the equation of the hyperbola where foci are $(0, \pm 12)$ and the length of the latus rectum is $36$.Preview
+Exercise 10.4i15 questions
  1. Q1Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola $\frac{x^2}{16} - \f…Free
  2. Q2Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola $\frac{y^2}{9} - \fr…Free
  3. Q3Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola $9y^2 - 4x^2 = 36$.Free
  4. Q4Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola $16x^2 - 9y^2 = 576$…Preview
  5. Q5Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola $5y^2 - 9x^2 = 36$.Preview
  6. Q6Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola $49y^2 - 16x^2 = 784…Preview
  7. Q7Find the equation of the hyperbola satisfying the given conditions: Vertices $(\pm 2, 0)$, foci $(\pm 3, 0)$.Preview
  8. Q8Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 5)$, foci $(0, \pm 8)$.Preview
  9. Q9Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 3)$, foci $(0, \pm 5)$.Preview
  10. Q10Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 5, 0)$, the transverse axis is of length $8$.Preview
  11. Q11Find the equation of the hyperbola satisfying the given conditions: Foci $(0, \pm 13)$, the conjugate axis is of length $24$.Preview
  12. Q12Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 3\sqrt{5}, 0)$, the latus rectum is of length $8$.Preview
  13. Q13Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 4, 0)$, the latus rectum is of length $12$.Preview
  14. Q14Find the equation of the hyperbola satisfying the given conditions: Vertices $(\pm 7, 0)$, $e = \frac{4}{3}$.Preview
  15. Q15Find the equation of the hyperbola satisfying the given conditions: Foci $(0, \pm \sqrt{10})$, passing through $(2, 3)$.Preview

Miscellaneous Examples

+Miscellaneous Examplesi3 questions
  1. Example 17The focus of a parabolic mirror as shown in Fig 10.31 is at a distance of $5$ cm from its vertex. If the mirror is $45$ cm deep, find the di…Free
  2. Example 18A beam is supported at its ends by supports which are $12$ metres apart. Since the load is concentrated at its centre, there is a deflection…Preview
  3. Example 19A rod $AB$ of length $15$ cm rests in between two coordinate axes in such a way that the end point $A$ lies on $x$-axis and end point $B$ li…Preview

Miscellaneous Exercise on Chapter 10

+Miscellaneous Exercisei8 questions
  1. Q1If a parabolic reflector is $20$ cm in diameter and $5$ cm deep, find the focus.Free
  2. Q2An arch is in the form of a parabola with its axis vertical. The arch is $10$ m high and $5$ m wide at the base. How wide is it $2$ m from t…Free
  3. Q3The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and $100$ m long is suppo…Free
  4. Q4An arch is in the form of a semi-ellipse. It is $8$ m wide and $2$ m high at the centre. Find the height of the arch at a point $1.5$ m from…Preview
  5. Q5A rod of length $12$ cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point $P$ on the r…Preview
  6. Q6Find the area of the triangle formed by the lines joining the vertex of the parabola $x^2 = 12y$ to the ends of its latus rectum.Preview
  7. Q7A man running a racecourse notes that the sum of the distances from the two flag posts from him is always $10$ m and the distance between th…Preview
  8. Q8An equilateral triangle is inscribed in the parabola $y^2 = 4ax$, where one vertex is at the vertex of the parabola. Find the length of the…Preview

Summary

- A conic section is the curve obtained by intersecting a double-napped right circular cone with a plane.

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 59 questions59 questions
  1. Q1Find the equation of the circle which touches the both axes in first quadrant and whose radius is $a$.Free
  2. Q2Show that the point $(x, y)$ given by $x = \dfrac{2at}{1+t^2}$ and $y = \dfrac{a(1-t^2)}{1+t^2}$ lies on a circle for all real values of $t$…Free
  3. Q3If a circle passes through the point $(0, 0)$, $(a, 0)$, $(0, b)$ then find the coordinates of its centre.Free
  4. Q4Find the equation of the circle which touches $x$-axis and whose centre is $(1, 2)$.Preview
  5. Q5If the lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ are tangents to a circle, then find the radius of the circle.Preview
  6. Q6Find the equation of a circle which touches both the axes and the line $3x - 4y + 8 = 0$ and lies in the third quadrant.Preview
  7. Q7If one end of a diameter of the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ is $(3, 4)$, then find the coordinate of the other end of the diameter…Preview
  8. Q8Find the equation of the circle having $(1, -2)$ as its centre and passing through $3x + y = 14$, $2x + 5y = 18$.Preview
  9. Q9If the line $y = \sqrt{3}\,x + k$ touches the circle $x^2 + y^2 = 16$, then find the value of $k$.Preview
  10. Q10Find the equation of a circle concentric with the circle $x^2 + y^2 - 6x + 12y + 15 = 0$ and has double of its area.Preview
  11. Q11If the latus rectum of an ellipse is equal to half of minor axis, then find its eccentricity.Preview
  12. Q12Given the ellipse with equation $9x^2 + 25y^2 = 225$, find the eccentricity and foci.Preview
  13. Q13If the eccentricity of an ellipse is $\dfrac{5}{8}$ and the distance between its foci is 10, then find latus rectum of the ellipse.Preview
  14. Q14Find the equation of ellipse whose eccentricity is $\dfrac{2}{3}$, latus rectum is 5 and the centre is $(0, 0)$.Preview
  15. Q15Find the distance between the directrices of the ellipse $\dfrac{x^2}{36} + \dfrac{y^2}{20} = 1$.Preview
  16. Q16Find the coordinates of a point on the parabola $y^2 = 8x$ whose focal distance is 4.Preview
  17. Q17Find the length of the line-segment joining the vertex of the parabola $y^2 = 4ax$ and a point on the parabola where the line-segment makes…Preview
  18. Q18If the points $(0, 4)$ and $(0, 2)$ are respectively the vertex and focus of a parabola, then find the equation of the parabola.Preview
  19. Q19If the line $y = mx + 1$ is tangent to the parabola $y^2 = 4x$ then find the value of $m$.Preview
  20. Q20If the distance between the foci of a hyperbola is 16 and its eccentricity is $\sqrt{2}$, then obtain the equation of the hyperbola.Preview
  21. Q21Find the eccentricity of the hyperbola $9y^2 - 4x^2 = 36$.Preview
  22. Q22Find the equation of the hyperbola with eccentricity $\dfrac{3}{2}$ and foci at $(\pm 2, 0)$.Preview
  23. Q23If the lines $2x - 3y = 5$ and $3x - 4y = 7$ are the diameters of a circle of area 154 square units, then obtain the equation of the circle.Preview
  24. Q24Find the equation of the circle which passes through the points $(2, 3)$ and $(4, 5)$ and the centre lies on the straight line $y - 4x + 3 =…Preview
  25. Q25Find the equation of a circle whose centre is $(3, -1)$ and which cuts off a chord of length 6 units on the line $2x - 5y + 18 = 0$.Preview
  26. Q26Find the equation of a circle of radius 5 which is touching another circle $x^2 + y^2 - 2x - 4y - 20 = 0$ at $(5, 5)$.Preview
  27. Q27Find the equation of a circle passing through the point $(7, 3)$ having radius 3 units and whose centre lies on the line $y = x - 1$.Preview
  28. Q28Find the equation of each of the following parabolas: (a) Directrix $x = 0$, focus at $(6, 0)$; (b) Vertex at $(0, 4)$, focus at $(0, 2)$; (…Preview
  29. Q29Find the equation of the set of all points the sum of whose distances from the points $(3, 0)$ and $(9, 0)$ is 12.Preview
  30. Q30Find the equation of the set of all points whose distance from $(0, 4)$ are $\dfrac{2}{3}$ of their distance from the line $y = 9$.Preview
  31. Q31Show that the set of all points such that the difference of their distances from $(4, 0)$ and $(-4, 0)$ is always equal to 2 represent a hyp…Preview
  32. Q32Find the equation of the hyperbola with: (a) Vertices $(\pm 5, 0)$, foci $(\pm 7, 0)$; (b) Vertices $(0, \pm 7)$, $e = \dfrac{4}{3}$; (c) Fo…Preview
  33. Q33The area of the circle centred at $(1, 2)$ and passing through $(4, 6)$ is (A) $5\pi$ (B) $10\pi$ (C) $25\pi$ (D) none of thesePreview
  34. Q34Equation of a circle which passes through $(3, 6)$ and touches the axes is (A) $x^2 + y^2 + 6x + 6y + 3 = 0$ (B) $x^2 + y^2 - 6x - 6y - 9 =…Preview
  35. Q35Equation of the circle with centre on the $y$-axis and passing through the origin and the point $(2, 3)$ is (A) $x^2 + y^2 + 13y = 0$ (B) $3…Preview
  36. Q36The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length $3a$ is…Preview
  37. Q37If the focus of a parabola is $(0, -3)$ and its directrix is $y = 3$, then its equation is (A) $x^2 = -12y$ (B) $x^2 = 12y$ (C) $y^2 = -12x$…Preview
  38. Q38If the parabola $y^2 = 4ax$ passes through the point $(3, 2)$, then the length of its latus rectum is (A) $\dfrac{2}{3}$ (B) $\dfrac{4}{3}$…Preview
  39. Q39If the vertex of the parabola is the point $(-3, 0)$ and the directrix is the line $x + 5 = 0$, then its equation is (A) $y^2 = 8(x + 3)$ (B…Preview
  40. Q40The equation of the ellipse whose focus is $(1, -1)$, the directrix the line $x - y - 3 = 0$ and eccentricity $\dfrac{1}{2}$ is (A) $7x^2 +…Preview
  41. Q41The length of the latus rectum of the ellipse $3x^2 + y^2 = 12$ is (A) $4$ (B) $3$ (C) $8$ (D) $\dfrac{4}{\sqrt{3}}$Preview
  42. Q42If $e$ is the eccentricity of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a < b)$, then (A) $b^2 = a^2(1 - e^2)$ (B) $a^2 = b^2(…Preview
  43. Q43The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half of the distance between the foci is (A) $\dfra…Preview
  44. Q44The distance between the foci of a hyperbola is 16 and its eccentricity is $\sqrt{2}$. Its equation is (A) $x^2 - y^2 = 32$ (B) $\dfrac{x^2}…Preview
  45. Q45Equation of the hyperbola with eccentricity $\dfrac{3}{2}$ and foci at $(\pm 2, 0)$ is (A) $\dfrac{x^2}{4} - \dfrac{y^2}{5} = \dfrac{4}{9}$…Preview
  46. Q46The equation of the circle having centre at $(3, -4)$ and touching the line $5x + 12y - 12 = 0$ is ________.Preview
  47. Q47The equation of the circle circumscribing the triangle whose sides are the lines $y = x + 2$, $3y = 4x$, $2y = 3x$ is ________.Preview
  48. Q48An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the length of the string an…Preview
  49. Q49The equation of the ellipse having foci $(0, 1)$, $(0, -1)$ and minor axis of length 1 is ________.Preview
  50. Q50The equation of the parabola having focus at $(-1, -2)$ and the directrix $x - 2y + 3 = 0$ is ________.Preview
  51. Q51The equation of the hyperbola with vertices at $(0, \pm 6)$ and eccentricity $\dfrac{5}{3}$ is ________ and its foci are ________.Preview
  52. Q52The line $x + 3y = 0$ is a diameter of the circle $x^2 + y^2 + 6x + 2y = 0$.Preview
  53. Q53The shortest distance from the point $(2, -7)$ to the circle $x^2 + y^2 - 14x - 10y - 151 = 0$ is equal to 5.Preview
  54. Q54If the line $lx + my = 1$ is a tangent to the circle $x^2 + y^2 = a^2$, then the point $(l, m)$ lies on a circle.Preview
  55. Q55The point $(1, 2)$ lies inside the circle $x^2 + y^2 - 2x + 6y + 1 = 0$.Preview
  56. Q56The line $lx + my + n = 0$ will touch the parabola $y^2 = 4ax$ if $ln = am^2$.Preview
  57. Q57If P is a point on the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1$ whose foci are S and S$'$, then $PS + PS' = 8$.Preview
  58. Q58The line $2x + 3y = 12$ touches the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 2$ at the point $(3, 2)$.Preview
  59. Q59The locus of the point of intersection of lines $\sqrt{3}x - y - 4\sqrt{3}k = 0$ and $\sqrt{3}kx + ky - 4\sqrt{3} = 0$ for different value o…Preview