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Subject

Mathematics

Chapter

Ch 7Conic Sections

Weightage~5marksapprox

7.1.1Double ConeRead →

7.1.2Conic Sections (the cross-sections of a double cone)Read →

7.1.3Definition of a Conic Section and EccentricityRead →

7.1.4Some Useful Terms of Conic SectionsRead →

7.1.5Standard Equation of a ParabolaRead →

7.1.6Tracing of the Parabola $y^2=4ax$Read →

7.1.7Focal Distance and Latus Rectum of a ParabolaRead →

7.1.8Other Standard Forms of a ParabolaRead →

7.1.9Parametric Form of a ParabolaRead →

7.1.10General (Shifted) Form of a ParabolaRead →

7.1.11Tangent to a ParabolaRead →

7.1.12Condition of Tangency for a ParabolaRead →

7.1.13Tangents from a Point to a ParabolaRead →

7.2Ellipse — Introduction33 Q

7.3Hyperbola — Introduction31 Q

7.4Summary: Conics ComparedRead →

More questions87 Q

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 5y² = 24x.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 y² = -20x.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 3x² = 8y.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 x² = -8y.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 3y² = -16x.Q6Find the equation of the parabola with vertex at the origin, axis along Y-axis and passing through the point (-10,-5).Q7Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (3,4).Q8Find the equation of the parabola whose vertex is O(0,0) and focus at (-7,0).Q9Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (1,-6).Q10Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (2,3).Q11For the parabola 3y² = 16x, find the parameter of the point (3,-4).Q12For the parabola 3y² = 16x, find the parameter of the point (27,-12).Q13Find the focal distance of a point on the parabola y² = 16x whose ordinate is 2 times the abscissa.Q14Find coordinate of the point on the parabola y²=12x whose parameter is 1/3. Also find focal distance.Q15Find coordinate of the point on the parabola 2y²=7x whose parameter is -2. Also find focal distance.Q16For the parabola y² = 4x, find the coordinate of the point whose focal distance is 17.Q17Find length of latus rectum of the parabola y² = 4ax passing through the point (2,-6).Q18Find the area of the triangle formed by the line joining the vertex of the parabola x² = 12y to the end points of latus rectum.Q19If a parabolic reflector is 20cm in diameter and 5 cm deep, find its focus.Q20Find coordinate of focus, vertex and equation of directrix and the axis of the parabola y = x² - 2x + 3.Q21Find the equation of tangent to the parabola y² = 12x from the point (2,5).Q22Find the equation of tangent to the parabola y² = 36x from the point (2,9).Q23If the tangent drawn from the point (-6,9) to the parabola y² = kx are perpendicular to each other, find k.Q24Two tangents to the parabola y² = 8x meet the tangents at the vertex in the point P and Q. If PQ = 4, prove that the equation of the locus of the point of intersection of two tangent is y² = 8(x + 2).Q25Find the equation of common tangent to the parabola y² = 4x and x² = 32y.Q26Find the equation of the locus of a point, the tangents from which to the parabola y² = 18x are such that some of their slopes is -3.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 5 meters above the road way at the centre of the bridge, find the length of the vertical supporting cable from the centre.Q28A circle whose centre is (4,-1) passes through the focus of the parabola x² + 16y = 0. Show that the circle touches the directrix of the parabola.Q95The line y = mx + 1 is tangent to the parabola y² = 4x if m is A) 1 B) 2 C) 3 D) 4 Q96The length of latus rectum of the parabola x² - 4x - 8y + 12 = 0 is A) 4 B) 6 C) 8 D) 10 Q97If the focus of the parabola is (0,-3) its directrix is y = 3 then its equation is A) x²=-12y B) x²=12y C) y²=12x D) y²=-12x Q98The coordinates of a point on the parabola y² = 8x whose focal distance is 4 are A) (1/2, ±2) B) (1, ±2√2) C) (2, ±4) D) none of these Q99The end points of latus rectum of the parabola y² = 24x are A) (6, ± 12) B) (12, ±6) C) (6, ± 6) D) none of these Q100Equation of the parabola with vertex at the origin and directrix x + 8 = 0 is A) y² = 8x B) y² = 32x C) y² = 16x D) x² = 32y Q101The area of the triangle formed by the line joining the vertex of the parabola x² = 12y to the end points of its latus rectum is A) 22 sq.units B) 20 sq.units C) 18 sq.units D) 14 sq.units Q102If P(dfracπ4) is any point on the ellipse 9x² + 25y² = 225. S and S' are its foci then SP · S'P = A) 13 B) 14 C) 17 D) 19 Q103The equation of the parabola having (2, 4) and (2, -4) as end points of its latus rectum is A) y² = 4x B) y² = 8x C) y² = -16x D) x² = 8y Q104If the parabola y² = 4ax passes through (3, 2) then the length of its latus rectum is A) 2/3 B) 4/3 C) 1/3 D) 4 Q105The eccentricity of rectangular hyperbola is A) 1/2 B) 1/√2 C) √2 D) 1/√3 Q106The equation of the ellipse having foci (± 4, 0) and eccentricity 1/3 is, A) 9x² + 16y² = 144 B) 144x² + 9y² = 1296 C) 128x² + 144y² = 18432 D) 144x² + 128y² = 18432 Q107The equation of the ellipse having eccentricity dfrac√32 and passing through (-√8, √3) is A) 4x² + y² = 4 B) x² + 4y² = 100 C) 4x² + y² = 100 D) x² + 4y² = 4 Q108If the line 4x - 3y + k = 0 touches the ellipse 5x² + 9y² = 45 then the value of k is A) ± 21 B) ±3√(21) C) ±3 D) ±3√(21)Q109The equation of the ellipse is 16x² + 25y² = 400. The equations of the tangents making an angle of 180° with the major axis are A) x = 4 B) y = ± 4 C) x = −4 D) x = ±5 Q110The equation of the tangent to the ellipse 4x² + 9y² = 36 which is perpendicular to the 3x + 4y = 17 is, A) y = 4x + 6 B) 3y + 4x = 6 C) 3y = 4x + 6√5 D) 3y = x + 25 Q111Eccentricity of the hyperbola 16x² - 3y² - 32x - 12y - 44 = 0 is A) √(17)/3 B) √(19)/3 C) √(19)/3 D) √(17)/3 Q112Centre of the ellipse 9x² + 5y² - 36x - 50y - 164 = 0 is at A) (2, 5) B) (1, −2) C) (−2, 1) D) (0, 0)Q113If the line 2x - y = 4 touches the hyperbola 4x² - 3y² = 24, the point of contact is A) (1, 2) B) (2, 3) C) (3, 2) D)(−2, −3)Q114The foci of hyperbola 4x² - 9y² - 36 = 0 are A) (±√(13), 0) B) (±√(11), 0) C) (±√(12), 0) D) (0, ±√(12))Q115For each of the following parabolas, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum: 2y² = 17x.Q116For each of the following parabolas, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum: 5x² = 24y.Q117Find the Cartesian co-ordinates of the points on the parabola y² = 12x whose parameter is 2.Q118Find the Cartesian co-ordinates of the points on the parabola y² = 12x whose parameter is -3.Q119Find the co-ordinates of a point of the parabola y² = 8x having focal distance 10.Q120Find the equation of the tangent to the parabola y² = 9x at the point (4, -6) on it.Q121Find the equation of the tangent to the parabola y² = 8x at t = 1 on it.Q122Find the equations of the tangents to the parabola y² = 9x through the point (4,10).Q123Show that the two tangents drawn to the parabola y² = 24x from the point (-6,9) are at the right angle.Q124Find the equation of the tangent to the parabola y² = 8x which is parallel to the line 2x + 2y + 5 = 0. Find its point of contact.Q125A line touches the circle x² + y² = 2 and the parabola y² = 8x. Show that its equation is y = ±(x+2).Q126Two tangents to the parabola y² = 8x meet the tangent at the vertex in P and Q. If PQ = 4, prove that the locus of the point of intersection of the two tangents is y² = 8(x + 2).Q127The slopes of the tangents drawn from P to the parabola y² = 4ax are m₁ and m₂. Using the quadratic satisfied by the slopes, state the two standard relations between m₁, m₂ and the coordinates of P(x₁,y₁).Q128The tangent at point P on the parabola y² = 4ax meets the y-axis in Q. If S is the focus, show that SP subtends a right angle at Q.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) Distance between foci (vi) distance between directrices of the curve: x²/25 + y²/9 = 1.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) Distance between foci (vi) distance between directrices of the curve: 16x² + 25y² = 400.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) Distance between foci (vi) distance between directrices of the curve: x²/144 - y²/25 = 1.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) Distance between foci (vi) distance between directrices of the curve: x² - y² = 16.Q133Find the equation of the ellipse in standard form if eccentricity = 3/8 and distance between its foci = 6.Q134Find the equation of the ellipse in standard form if the length of major axis 10 and the distance between foci is 8.Q135Find the equation of the ellipse in standard form if passing through the points (-3, 1) and (2, -2).Q136Find the eccentricity of an ellipse if the distance between its directrices is three times the distance between its foci.Q137For the hyperbola x²/100 - y²/25 = 1, prove that SA · S'A = 25, where S and S' are the foci and A is the vertex.Q138Find the equation of the tangent to the ellipse x²/5 + y²/4 = 1 passing through the point (2,-2).Q139Find the equation of the tangent to the ellipse x² + 4y² = 100 at (8,3).Q140Show that the line 8y + x = 17 touches the ellipse x² + 4y² = 17. Find the point of contact.Q140Show that the line 8y + x = 17 touches the ellipse x² + 4y² = 17. Find the point of contact.Q141Tangents are drawn through a point P to the ellipse 4x² + 5y² = 20 having inclinations θ₁ and θ₂ such that tanθ₁ + tanθ₂ = 2. Find the equation of the locus of P.Q141Tangents are drawn through a point P to the ellipse 4x² + 5y² = 20 having inclinations θ₁ and θ₂ such that tanθ₁ + tanθ₂ = 2. Find the equation of the locus of P.Q142Show that the product of the lengths of its perpendicular segments drawn from the foci to any tangent line to the ellipse x²/25 + y²/16 = 1 is equal to 16.Q143Find the equation of the hyperbola in the standard form if Length of conjugate axis is 5 and distance between foci is 13.Q144Find the equation of the hyperbola in the standard form if eccentricity is 3/2 and distance between foci is 12.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.Q146Find the equation of the tangent to the hyperbola 7x² - 3y² = 51 at (-3, -2).Q147Find the equation of the tangent to the hyperbola x = 3secθ,y = 5tanθ at θ = π/3.Q148Find the equation of the tangent to the hyperbola x²/25 - y²/16 = 1 at P(30°).Q149Show that the line 2x - y = 4 touches the hyperbola 4x² - 3y² = 24. Find the point of contact.Q150Find the equations of the tangents to the hyperbola 3x² - y² = 48 which are perpendicular to the line x + 2y - 7 = 0.Q151Two tangents to the hyperbola make angles θ₁, θ₂, with the transverse axis. Find the locus of their point of intersection if tanθ₁ + tanθ₂ = k.

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Ch 7 — Conic Sections

Miscellaneous Exercise 7(II) · Q130

Q. Find the

(i) lengths of the principal axes

(ii) co-ordinates of the foci

(iii) equations of directrices

(iv) length of the latus rectum

(v) Distance between foci

(vi) distance between directrices of the curve:

16x^2 + 25y^2 = 400

.

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Concept understanding — Focal Properties and Latus Rectum of an Ellipse

Several powerful derived facts follow from the ellipse's standard equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ( $b^2=a^2(1-e^2)$ , $a>b$ ):

The latus rectum — the focal chord perpendicular to the major axis — has length $\dfrac{2b^2}{a}$ , with end points $\left(ae,\pm\dfrac{b^2}{a}\right)$ .
The distance between the two directrices is $\dfrac{2a}{e}$ . …

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