HOTs Conic Sections Class 11 Mathematics

Question. The equation of an ellipse whose focus is (–1, 1), whose directrix is x − y + 3 = 0 and whose eccentricity is 1/2 is given by:
(a) 7x² + 2xy + 7y² +10x −10y + 7 = 0
(b) 7x² − 2xy + 7 y² −10x +10y + 7 = 0
(c) 7x² − 2xy + 7y² −10x −10y − 7 = 0
(d) 7x² − 2xy + 7 y² +10x +10y − 7 = 0

Question. The equation of parabola whose focus is (5, 3) and directrix is 3x − 4y +1= 0, is:
(a) (4x + 3y)² − 256x −142y + 849 = 0
(b) (4x − 3y)² − 256x −142y + 849 = 0
(c) (3x + 4y)² −142x − 256y + 849 = 0
(d) (3x − 4y)² − 256x −142y + 849 = 0

Question. If the parabola 2 y = 4ax passes through (–3, 2), then length of its latus rectum is:
(a) 2/3
(b) 1/3
(c) 4/3
(d) 4

Question. The number of values of ‘c’ such that the straight line y = 4x + c touches the curve x²/4 + y² =1is :
(a) 0
(b) 1
(c) 2
(d) Infinite

Question. If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then:
(a) a² (CG)² + b² (Cg) = (a² − b² )²
(b) a² (CG)² − b² (Cg) = (a² − b² )²
(c) a² (CG)² − b² (Cg) = (a² + b² )²
(d) None of these

Question. x − 2 = t² , y = 2t are the parametric equations of the parabola:
(a) y² = 4x
(b) y² = −4x
(c) x²  = −4y
(d) y²  = 4(x − 2)

Question. The straight line y = 2x +λ does not meet the parabola y² = 2x, if:
(a) λ <1/4
(b) λ >1/4
(c) λ = 4
(d) λ =1

Question. If x + y = k is a normal to the parabola y² = 12x, then k is:
(a) 3
(b) 9
(c) –9
(d) –3

Question. The normals at three points P, Q, R of the parabola y² = 4ax meet in (h, k), the centroid of triangle PQR lies on:
(a) x = 0
(b) y = 0
(c) x = −a
(d) y = a

Question. If m₁ and m₂ are the slopes of the tangents to the hyperbola x²/25 – y²/16 = 1 which pass through the point (6,2), then:     
(a) m₁+m₂ = 24/11
(b) m₁.m₂ = 20/11
(c) m₁+m₂ = 48/11
(d) m₁.m₂ = 11/20

Question. If the points (au² ,2au) and (av² ,2av) are the extremities of a focal chord of the parabola y² = 4ax, then:
(a) uv −1 = 0
(b) uv +1 = 0
(c) u + v = 0
(d) u −v = 0

Question. Equation of diameter of parabola y² = x corresponding to the chord x − y +1 = 0 is:
(a) 2y = 3
(b) 2y =1
(c) 2y = 5
(d) y = 1

Question. The length of the sub-tangent to the parabola y² = 16x at the point, whose abscissa is 4, is:
(a) 2
(b) 4
(c) 8
(d) None of these

Question. The pole of the line 2x = y with respect to the parabola y² = 2x is:
(a) (0 , 1/2)
(b) (1/2 , 0) 
(c) (0 , –1/2)
(d) None of these 

Question. A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is ( y − 2)² = 4(x +1). after reflection, the ray must pass through the point:
(a) (0, 2)
(b) (2, 0)
(c) (0, –2)
(d) (–1, 2)

Question. If the normal at (ct , c/t) on the curve xy = c² meets the curve again in t′, then:
(a) t’ = -1/t₃
(b)  t’ = -1/t
(c)  t’ = -1/t₂
(d)  t₂ = -1/t₂

Question. If a circle cuts a rectangular hyperbola xy² = c in A, B, C, D and the parameters of these four points be t₁ , t₂ , t₃ and t₄ respectively. Then:
(a) t₁t₂ = t₃t₄
(b)  t₁t₂t₃t₄ = 1
(c)  t₁ = t₂ 
(d)  t₃ = t₄

Question. If 1 P(x, y),F = (3,0), 2 F = (−3,0) and 16x² + 25y² = 400, then PF₁ + PF₂ equals:
(a) 8
(b) 6
(c) 10
(d) 12

Question. The equation x² − 2xy + y² + 3x + 2 = 0 represents: 
(a) A parabola
(b) An ellipse
(c) A hyperbola
(d) A circle

Question. The equation of a directrix of the ellipse x²/16 + y²/25 = 1 is :
(a) y = 25/3
(b) x = 3
(c) x = −3
(d) x = 3/25

Question. If the tangent to the parabola y² = ax makes an angle of 45º with x-axis, then the point of contact is:
(a) (a/2 , a/2)
(b) (a/4 , a/4)
(c) (a/2 , a/4)
(d) (a/4 , a/2)

Question. The line x − y + 2 = 0 touches the parabola y² = 8x at the point:
(a) (2,–4)
(b) (1,2 √2)
(c) (4,−4 √2)
(d) (2, 4)

Question. The distance of the point ‘θ ‘ on the ellipse x²/9 + y²/4 = 1 from a focus is:
(a) a(e + cosθ )
(b) a(e − cosθ )
(c) a(1+ e cosθ )
(d) a(1+ 2ecosθ )

Question. The centre of 14x² − 4xy +11y² − 44x − 58y + 71 = 0 is:
(a) (2, 3)
(b) (2, –3)
(c) (–2, 3)
(d) (–2, –3)

Question. Let E be the ellipse x²/9 + y²/4 = 1 and C be the circle x² + y² = 9. Let P and Q be the points (1, 2) and (2, 1) respectively. Then:
(a) Q lies inside C but outside E
(b) Q lies outside both C and E
(c) P lies inside both C and E
(d) P lies inside C but outside E

Question. What will be the equation of the chord of contact of tangents drawn from (3, 2) to the ellipse x² + 4y² = 9 ?
(a) 3x + 8y = 9
(b) 3x + 8y = 25
(c) 3x + 4y = 9
(d) 3x + 8y + 9 = 0

Question. The pole of the straight line x + 4y = 4 with respect to ellipse x² + 4y² = 4 is:
(a) (1, 4)
(b) (1, 1)
(c) (4, 1)
(d) (4, 4)

Question. If one end of a diameter of the ellipse 4x² + y² = 16 is ( √3, 2), then the other end is:
(a) (− √3, 2)
(b) ( √3,− 2)
(c) (− √3,− 2)
(d) (0,0)

Question. If θ and φ are eccentric angles of the ends of a pair of conjugate diameters of the ellipse x²/a² + y²/b² = 1,then θ −φ is equal to:
(a) ± π/2
(b) ± π
(c) 0
(d) None of these

Question. In the ellipse 25x² + 9y² + 150x – 190y + 225 = 0 ?       
(a) foci are at (3,1), (3, 9)
(b) e = 4/5
(c) centre is (5,3)
(d) major axis is

Question. Length of sub-tangent and subnormal at the point (Image 30)       

Question. The equation of a parabola is 2 y = 4x. P(1,3) and Q(1,1) are two points in the xy-plane. Then, for the parabola:
(a) P and Q are exterior points
(b) P is an interior point while Q is an exterior point
(c) P and Q are interior points
(d) P is an exterior point while Q is an interior point

Question. The equation of the conic with focus at (1, – 1), directrix along x − y +1 = 0 and with eccentricity 2 is:
(a) x² − y² =1
(b) xy =1
(c) 2xy + 4x − 4y −1 = 0
(d) 2xy + 4x − 4y −1 = 0

Question. The locus of the point of intersection of tangents to the hyperbola 4x² − 9y² = 36 which meet at a constant angle π / 4, is:
(a) (x² + y² − 5)² = 4(9y² − 4x² + 36)
(b) (x² + y² − 5)² = 4(9y² − 4x² + 36)
(c) 4(x² + y² − 5)² = (9y² − 4x² + 36)
(d) None of these

Question. The equation of the normal to the hyperbola  x²/16 – y²/9 = 1 at the point (8,3√3) is
(a) √3x + 2y = 25
(b) x + y = 25
(c) y + 2x = 25
(d) 2x + √3y = 25

Question. If the normal at ‘φ ‘ on the hyperbola x²/a² + y²/b² = 1 meets transverse axis at G, then AG.A’G = ? (Where A and A’ are the vertices of the hyperbola)
(a) a² (e4 sec2 φ −1)
(b) (a2 e4 sec2 φ −1)
(c) a2 (1− e4 sec2 φ )
(d) None of these

Question. The equation of the chord of contact of tangents drawn from a point (2, –1) to the hyperbola 16x² − 9y² =144 is:
(a) 32x + 9y = 144
(b) 32x + 9y = 55
(c) 32x + 9y +144 = 0
(d) 32x + 9y + 55 = 0

Question. The point of intersection of tangents drawn to the hyperbola x²/a² + y²/b² = 1 at the points where it is intersected by the line lx + my + n = 0 is:  (Image 38)         

Question. If the polar of a point w.r.t. x²/a² + y²/b² = 1 touches the hyperbola x²/a² + y²/b² = 1 then the locus of the point is:
(a) Given hyperbola
(b) Ellipse
(c) Circle
(d) None of these

Question. If a pair of conjugate diameters meet the hyperbola and its conjugate in P and D respectively, then CP² −CD² = ?
(a) a² + b²
(b) a² − b²
(c) a²/b²
(d) None of these

Question. If the tangent at the point (asecα , b tanα ) to the hyperbola x²/a² – y²/b² = meets the transverse axis at T, then the distance of T form a focus of the hyperbola is:         
(a) a(e − cosα )
(b) b(e + cosα )
(c) a(e + cosα )
(d) √(a² e² + b² + cot² α)

Question. From any point on the hyperbola,  x²/a² + y²/b² = 1 tangents are drawn to the hyperbola x²/a² + y²/b² = 2 The area cut-off by the chord of contact on the asymptotes is equal to:
(a) ab/2
(b) ab
(c) 2ab
(d) 4ab

Question. The area of the quadrilateral formed by the tangents at the end points of latus- rectum to the ellipse x²/9 + y²/5 = 1 is :
(a) 27/4 sq. units
(b) 9 sq. units
(c) 27/2 sq. units
(d) 27sq. units

Question. The equation of normal at the point (0, 3) of the ellipse 9x² + 5y² = 45 is:
(a) y −3 = 0
(b) y + 3 = 0
(c) x-axis
(d) y-axis

Question. The number of tangents to the hyperbola x²/4 – y²/3 = 1 through (4, 1) is:
(a) 1
(b) 2
(c) 0
(d) 3

Question. The points of contact of the line i y = x −1 with 3x² − 4y² = 12 is:
(a) (4, 3)
(b) (3, 4)
(c) (4,–3)
(d) None of these

Question. The combined equation of the asymptotes of the hyperbola 2x² + 5xy + 2y² + 4x + 5y = 0 ?
(a) 2x² + 5xy + 2y² = 0
(b) 2x² + 5xy + 2y² − 4x + 5y + 2 = 0 = 0
(c) 2x² + 5xy + 2y² + 4x + 5y − 2 = 0
(d) 2x² + 5xy + 2y² + 4x + 5y + 2 = 0

Question. If 5x² +λy² = 20 represents a rectangular hyperbola, then λ equals:
(a) 5
(b) 4
(c) – 5
(d) None of these

Question. Equation of common tangent of y = x², y = – x² + 4x – 4 is:                    
(a) y = 4 (x – 1)
(b) y = 0
(c) y = – 4 (x – 1)
(d) y = – 30 x – 50

Question. Let P(x₁,y₁) and Q(x₂,y₂), y₁ < 0, y₂ < 0, be the end points of the latus rectum of the ellipse x² + 4y² = 4.The equations of parabolas with latus rectum PQ are:         
(a) x² + 2√3y = 3+ √3
(b)x² − 2√3y = 3+ √3
(c) x² + 2√3y = 3− √3
(d) x² − 2√3y = 3− √3