Class 11 Mathematics Conic Sections MCQs Set 07

Question: If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is :
(a) (x² + y²)² = 4R²x²y²
(b) (x² + y²)³ = 4R²x²y²
(c) (x² + y²)² = 4Rx²y²
(d) (x² + y²)² = 4R²xy
Answer: b

Question: The equation of the ellipse with focus at (± 5, 0) and x = 36/5 as one directrix is
(a) x²/36 + y²/25 = 1
(b) x²/36 + y²/11 = 1
(c) x²/25 + y²/11 = 1
(d) None of these
Answer: b

Question: The locus of the middle points of the focal chords of parabola y²= 4ax is
(a) y² = a(x-a)
(b) y² = 2a(x-a)
(c) y² = 4a(x-a) 
(d) None of these 
Answer: b

Question: If a line, y = mx + c is a tangent to the circle, (x – 3)² + y² = 1 and it is perpendicular to a line L₁, where L₁ is the tangent to the circle, x² + y² = 1 at the point (1/√2, 1/√2) ; then :
(a) c² – 7c + 6 = 0
(b) c² + 7c + 6 = 0
(c) c² + 6c + 7 = 0
(d) c² – 6c + 7 = 0
Answer: c

Question: A tangent to a parabola y² = 4ax is inclined at π/3 with the axis of the parabola. The point of contact is
(a) (a/3,- 2a/√3)
 (b) (3a,-2√3a)
(c) (3a,2√3a)
(d) None of these 
Answer: a

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) 
Answer: d

Question: If the normal to the parabola y² = 4ax at the point P (at²,2at) cuts the parabola again at Q(aT², 2aT), then
(a) -2≤ t ≤ 2
(b) t ∈(-∞,-8) ∪ (8,∞)
(c) T2 <8
(d) T2 ≥8 
Answer: d

Question: Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then:
(a) 1/√a = 1/√b + 1/√c
(b) 1/√b = 1/√a + 1/√c
(c) a, b, c are in A.P
(d) √a, √b, √c are in A.P.
Answer: a

Question: Let C₁ and C₂ be the centres of the circles x² + y² – 2x –2y – 2 = 0 and x² + y² – 6x –6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles then, the area (in sq. units) of the quadrilateral PC₁QC₂ is :
(a) 8
(b) 6
(c) 9
(d) 4
Answer: d

 Question: The parametric equation of a parabola is x = t²+1, y=2t+1.
The cartesian equation of its directrix is
(a) x = 0
(b) x + = 1 0
(c) y = 0
(d) None of the above
Answer: a

Question: Consider an ellipse, whose centre is at the origin and its ma or axis is along the x–axis. If its eccentricity is 3/5 and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is :
(a) 8
(b) 32
(c) 80
(d) 40
Answer: d

Question: A square is inscribed in the circle x² + y² – 6x + 8y -103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is :
(a) 6
(b) √137
(c) √41
(d) 13
Answer: c

Question: If a circle C passing through the point (4, 0) touches the circle x² + y² + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is:
(a) 2√5
(b) 4
(c) 5
(d) √57
Answer: c

Question: The length of the latusrectum of the parabola
169 {(x-1)² + (y-3)²} = (5x-12y+17)² is
(a) 14/13
(b) 12/13
(c) 28/13
(d) None of these 
Answer: c

Question: The position of the point (– 2, 2) with respect to the parabola y²-4y +9x+13 =0 is
(a) inside
(b) outside
(c) on
(d) None of these 
Answer: a

Question: The angle of intersection between the curves x² = 4(y+1) and x² = -4(y+1) is
(a) π/6
(b) π/4
(c) 0
(d) π/2 
Answer: a

Question: Which one of the following points lies outside the ellipse (x²/ a²) + (y²/ b²) = 1 ?
(a) (a, 0)
(b) (0, b)
(c) (– a, 0)
(d) (a, b)
Answer: d

Question: The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points (4, –1) and (–2, 2) is :
(a) 1/2
(b) 2/√5
(c) √3/2
(d) √3/4
Answer: c

Question: Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (–3, 1) and has eccentricity √2/5/is
(a) 5x² + 3y² – 48 = 0
(b) 3x² + 5y² – 15 = 0
(c) 5x² + 3y² – 32 = 0
(d) 3x² + 5y² – 32 = 0
Answer: d

Question: Length of the latus rectum of the ellipse x²/a² + y²/b² = 1 is
(a) b²/a²
(b) 2b/a
(c) 2b²/a
(d) 2a²/b
Answer: c

Question: The tangents and normal at the ends of the latusrectum of a parabola form a
(a) cyclic quadrilateral
(b) rectangle
(c) square
(d) None of these 
Answer: c

Question: If tangents at A and B on the parabola y² = 4ax intersect at point C, then ordinates of A C, and B are
(a) always in AP
(b) always in GP
(c) always in HP
(d) None of these 
Answer: a

Question: The length of latusrectum of the parabola 169 {(x-1)² +(y-3)²}= (5x-12y+17)² is
(a) 14/13
(b) 28/13
(c) 12/13
(d) None of these 
Answer: b

Question: The length of the semi-latus rectum of an ellipse is one third of its major axis, its eccentricity would be
(a) 2/3
(b) √2/3
(c) 1/√3
(d) 1/√2
Answer: c

Question: The largest value of r for which the region represented by the set {ω ∈ C|ω – 4 – i| ≤r} is contained in the region represented by the set (z ∈c / | z -1|≤| z + i |), is equal to:
(a) (5/2)2
(b) 2√2
(c) (3/2)√2
(d) √17
Answer: a

Question: The circle x²+ y² =5 meets the parabola y² = 4x at p and Q Then, the length P and Q is equal to
(a) 2
(b) 2 2
(c) 4
(d) None of these
Answer: c

Question: The equation of the hyperbola whose vertices are (± 2, 0) and foci are (± 3, 0) is x²/a² – y²/b² = 1. Sum of a² and b² is
(a) 5
(b) 4
(c) 9
(d) 1
Answer: c

Question: Two circles S₁ = x² + y² + 2g₁x + 2f₁y + c₁ = 0 and S₂ = x² + y² + 2g₂x + 2f₂y + c₂ = 0 cut each other orthogonally, then :
(a) 2g₁g₂ + 2f₁f₂ = c₁ + c₂
(b) 2g₁g₂ – 2f₁f₂ = c₁ + c₂
(c) 2g₁g₂ + 2f₁f₂ = c₁ – c₂
(d) 2g₁g₂ – 2f₁f₂ = c₁ – c₂
Answer: a

Question: If a circle passes through the point (a, b) and cuts the circle x² + y² = 4orthogonally, then the locus of its centre is
(a) 2ax – 2by – (a² + b² + 4) = 0
(b) 2ax + 2by – (a² + b² + 4) = 0
(c) 2ax – 2by + (a² + b² + 4) = 0
(d) 2ax + 2by + (a² + b² + 4) = 0
Answer: b

Question: If tangents are drawn to the ellipse x² + 2y² = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :
(a) 1/4x² + 1/2y² = 1
(b) x²/4 + y²/2 = 1
(c) 1/2x² + 1/4y² = 1
(d) x²/2 + y²/4 = 1
Answer: c

Question: The focus of the parabola y²=4y-4x is
(a) (0, 2)
(b) (1, 2)
(c) (2, 0)
(d) (2, 1) 
Answer: a

Question: The equation of parabola having vertex (0, 0) passing through (2, 3) and axis is along x-axis is 
(a) x² = 9/2y 
(b) y² = 9/2 x 
(c) y² = – 9/2x 
(d) x² = -9/2 y
Answer: b

Question: Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, x²/4 + y²/2 = 1 from any of its foci?
(a) (-2, √3)
(b) (-1, √2)
(c) (-1, √3)
(d) (1, 2)
Answer: c

Question: If a circle C passing through (4, 0) touches the circle x2 + y² + 4x – 6y – 12 = 0 externally at a point (1, –1), then the radius of the circle C is :
(a) 5
(b) 2√5
(c) 4
(d) √57
Answer: a

Question: The equation of parabola having vertex (0,0), passing through (5, 2) and symmetric with respect to y-axis is
(a) 3x² =25y 
(b) 2x² = 25y 
(c) 2y² = 25y 
(d) None of these 
Answer: b

Question: If the parabola y²=4ax passes through the point (3, 2) then the length of its latusrectum is
(a) 2/3
(b) 4/3
(c) 1/3
(d) 4 
Answer: b

Question: If the point P( 4,2 ) is one end of the focal chord PQ of the parabola y2=x, then the slope of the tangent at Q is
(a) -1/4
(b) 1/4
(c) 4
(d) -4
Answer: c

Question: The equation of the parabola having vertex at the origin axis on the y-axis and passing through the point( , ) 6 3 – is
(a) y²=12x +6 
(b) x²= 12y
(c) x² = 12y 
(d) y² = 12x+6 
Answer: c

Question: If e₁ is the eccentricity of the ellipse x²/16 + y²/25 = 1 and e₂ is the eccentricity of the hyperbola passing through the foci of the ellipse and e₁e₂ = 1, then equation of the hyperbola is :
(a) x²/9 − y² /16 = 1
(b) x²/16 − y² /9 = 1
(c) x²/9 − y² /25 = 1
(d) x²/9 − y² /36 = 1
Answer: b

Question: If the eccentricity and length of latus rectum of a hyperbola are √13/3 and 10/3 units respectively, then what is the length of the transverse axis?
(a)7/2 unit
(b) 12 unit
(c) 15/2 unit
(d) 15/4 unit
Answer: c

Question: The coordinates of a point on the parabola y²=8x whose focal distance is 4, is 
(a) (– 2, – 4), (2, – 4)
(b) (2, 4), (2, – 4)
(c) (1, 4), (2, – 4) (d)
(2, – 4), (– 2, 4) 
Answer: b