Class 11 Mathematics Conic Sections MCQs Set 05

Question: Find the equation of the ellipse whose foci are (2, 3), (–2, 3) and whose semi-minor axis is of length 5.
(a) 5x² + 9y² + 54y + 36 = 0
(b) 5x² + 9y² - 54y + 36 = 0
(c) 5x² + 9y² - 54y - 36 = 0
(d) None of these
Answer: b

Question: The eccentric angle of a point on the ellipse x²/6 + y²/2 = 1 whose distance from the centre of the ellipse is 2, is
(a) π/4
(b) 3π/2
(c) 5π/3
(d) 7π/6
Answer: a

Question: If e is eccentricity of ellipse x²/a² + y²/b² = 1 (a > b) and e' is eccentricity of x²/a² + y²/b² = 1 (a < b) , then
(a) e = e'
(b) ee' = 1
(c) 1/e² + 1/(e')² = 1
(d) None of these
Answer: c

Question: If the normals atP(q)and Q(π/2 +θ) to the ellipse x²/a² + y²/b² = 1 meet the major axis at Gand grespectivel,y the PG² + Qg² is equal to
(a) b²(1- e²)(2 - e²)
(b) b²(e⁴- e² + 2)
(c) a2(1+ e²)(2 + e²)
(d) b²(1+ e²)(2 + e²)
Answer: b

Question: The angle between the pair of tangents drawn from the point (1, 2) to the ellipse 3x² + 2y² = 5 , is
(a) tan^-1 (12 /5)
(b) tan^-1 (6/ √5)
(c) tan^-1(2 /√5)
(d) tan^-1 (6 /5)
Answer: c

Question: A (1/√2,1/√2) is a point on the circle x²+y²=1 and B is another point on the circle such that arc length AB =π/2 units. Then, coordinates of B can be
(a) (1/√2,-1/2) 
(b) (-1/√2,1/√2
(c) (−1/√2,-1/√2) 
(d) None of these
Answer: a,b

Question: If e is the eccentricity of the ellipse x²/a² + y²/b² = 1 (a < b), then
(a) b² = a² (1 -e²)
(b) a² = b²(1 -e²)
(c) a² = b² = (e² - 1)
(d) b² = a² = (e² - 1)
Answer: b

Question: The equation of the ellipse whose foci are (±2, 0) and eccentricity 1/2, is
(a) x²/12 + y²/16 = 1
(b) x²/16 + y²/12 = 1
(c) x²/16 + y²/8 = 1
(d) None of these
Answer: b

Question: On the ellipse 4x² + 9y² = 1, the point at which the tangent is parallel to the line 8x = 9y, is
(a) (2/5 , 1/5)
(b) (-2/5 , 1/5)
(c) (-2/5 , -1/5)
(d) None of these
Answer: b

Question: Find the equation of an ellipse, if major axis on the x-axis and passes through the points (4, 3) and (6, 2).
(a) x²/13+ y²/52= 1
(b) x²/40+ y²/10= 1
(c) x²/52+ y²/13= 1
(d) None of these
Answer: c

Question: The curve with parametric equations x = α + 5cos q, y = β + 4 sin q (where, q is parameter) is
(a) a parabola
(b) an ellipse
(c) a hyperbola
(d) None of these
Answer: b

Question: The curve represented by the equation 4x² + 16y² - 24x - 32y - 12 = 0 is
(a) a parabola
(b) a pair of straight lines
(c) an ellipse with eccentricity 1/2
(d) an ellipse with eccentricity √3/2
Answer: d

Question: In an ellipse length of minor axis is 8 and eccentricity is √5/3. The length of major axis is
(a) 6
(b) 12
(c) 10
(d) 16
Answer: b

Question: In an ellipse the distance between the foci is 8 and the distance between the directrices is 25. The length of major axis is
(a) 10√2
(b) 20√2
(c) 30√2
(d) None of these
Answer: a

Question: The line x = at² meets the ellipse x²/a² + y²/b² = 1 in the real points, if
(a) |t | < 2
(b) | t | ∈ 1
(c) |t | > 1
(d) None of these
Answer: b

Question: The equation of the chord of the circle x² + y² = a having (x1 , y1 ) as its mid-point is: 
(a) xy₁ + yx₁ = a²
(b) x₁ + y₁ = a
(c) xx₁ + yy₁ = x₁² + y₁²
(d) xx₁ + yy₁ = a².
Answer: c

Question: If x cos α + y sin α = pis a tangent to the ellipse, then
(a) a² sin α +b² cosα = p²
(b) a² + b²sin²α = p² cosec²α
(c) a² + cos²α = b²sin²α = p²
(d) None of the above
Answer: c

Question: Number of tangents from (7, 6) to ellipse x²/16 + y²/25 = 1 is
(a) 0
(b) 1
(c) 2
(d) None of these
Answer: c

Question: The distances from the foci of P(x , y ) 1 1 on the ellipse x²/9 + y²/25 = 1 are
(a) 4 ± 5/4 y1
(b) 5 ± 4/5 x1
(c) 5 ± 4/5 y1
(d) None of these
Answer: c

Question: If the latusrectum of an ellipse is equal to half of minor axis, then its eccentricity is
(a) √15/3
(b) √15/2
(c) √15/6
(d) √15/4
Answer: d

Question: If the eccentricity of an ellipse is 5/8 and the distance between its foci is 10, then length of latus rectum of the ellipse is
(a) 39/7
(b) 39/4
(c) 39/5
(d) 39/8
Answer: b

Question: A tangent at any point to the ellipse 4x² + 9y² = 36 is cut by the tangent at the extremities of the major axis at T and T'. The circle on T T' as diameter passes through the point
(a) (0, √5 )
(b) ( √5, 0)
(c) (2, 1)
(d) (0, - √5 )
Answer: b

Question: If vertices and foci of an ellipse are (0, ± 13) and (0, ± 5) respectively, then the equation of an ellipse is
(a) x²/144 + y²/169 = 1
(b) x²/169 + y²/144 = 1
(c) x²/12+ y²/13= 1
(d) None of these
Answer: a

Question: The sum of focal distance of any point on the ellipse with major and minor axes as 2a and 2b respectively, is equal to
(a) 2a
(b) 2 a/b
(c) 2 b/a
(d) b/a
Answer: a

Question: If the line x + 2by + 7 = 0 is a diameter of the circle x² + y² − 6x + 2y = 0, then b = ?
(a) 3
(b) –5
(c) –1
(d) 5
Answer: b

Question: An ellipse is described by using endles string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and distance between the pins respectively in cm, are
(a) 6, 2√5
(b) 6, √5
(c) 4, 2√5
(d) None ofthese
Answer: d

Question: The locus of the centre of the circle which cuts off intercepts of length 2a and 2b from x-axis and y-axis respectively, is:
(a) x + y = a + b
(b) x² + y² = a² + b²
(c) x² − y² = a² − b²
(d) x² + y² = a² − b²
Answer: c

Question: The equations to the tangents to the circle x² + y² − 6x + 4y =12 which are parallel to the straight line 4x+3y+5=0, are:
(a) 3x − 4y −19 = 0, 3x − 4y + 31 = 0
(b) 4x + 3y −19 = 0, 4x + 3y + 31 = 0
(c) 4x + 3y +19 = 0, 4x + 3y − 31 = 0
(d) 3x − 4y +19 = 0, 3x − 4y + 31 = 0
Answer: c

Question: If the eccentricity of the two ellipse x²/169 + y²/25 = 1 and x²/a² + y²/b² = 1 are equal, then the value of a/b is
(a) 5/13
(b) 6/13
(c) 13/5
(d) 13/6
Answer: c

Question: If Pis a point on the ellipse x²/16 + y²/25 = 1 1whose foci are S and S', then PS + PS' is equal to
(a) 8
(b) 7
(c) 5
(d) 10
Answer: d

Question: If the distances from the origin to the centres of three circles x² + y² + 2λ₁x −C² = 0 ( 1,2,3) i x + y + λ x − c = i = are in G.P. then the lengths of the tangents drawn to them from any point on the circle x² + y² = c² are in:
(a) (a)P.
(b) G.P.
(c) H.P.
(d) None of these
Answer: b

Question: If Qbe the point on the auxiliary circle corresponding to a point P on an ellipse. Then, the normals at P and Q meet on
(a) a fixed circle
(b) an ellipse
(c) a hyperbola
(d) None of these
Answer: a

Question: Find the distance between the directrices of the ellipse x²/36 + y²/20 = 1.
(a) -18
(b) 18
(c) 17
(d) 19
Answer: b

Question: The angle between a pair of tangents drawn from a point P to the circle x² + y² + 4x − 6 y + 9sin α +13cos α = 0 is 2α The equation of the locus of the point P is:
(a) x² + y² + 4x − 6y + 4 = 0
(b) x² + y² + 4x − 6y − 9 = 0
(c) x² + y² + 4x − 6y − 4 = 0
(d) x² + y² + 4x − 6y + 9 = 0
Answer: d

Question: The line lx + my + n = 0 is a normal to the circle x² + y² + 2gx + 2 fy + c = 0, if:
(a) lg+ mf − n = 0
(b) lg+ mf + n = 0
(c) lg−mf − n = 0
(d) lg−mf + n = 0
Answer: a

Question: If the circle x² + y² +2gx+2fy+c =0cuts each of the circle x² + y² −4 = 0, x² + y² −6x−8y+10=0 and x² + y² +2x−4y−2=0 at the extremities of a diameter, then
(a) c = −4
(b) g + f = c −1
(c) g² + f² − c =17
(d) g f = 6
Answer: All

Question: Equation of the circle having diameter x − 2 y + 3 = 0, 4 x − 3 y + 2 = 0 and radius equal to 1 is
(a) (x −1)² + ( y − 2)² =1
(b) (x − 2)² + ( y −1)² =1
(c) x² + y² − 2x − 4y + 4 = 0
(d) x² + y² −3x − 4y + 7 = 0
Answer: a,b

Question: The common chord of the circle x² + y² + 4x +1 = 0 and x² + y² + 6x + 2y + 3 = 0 is:
(a) x + y +1 = 0
(b) 5x + y + 2 = 0
(c) 2x + 2y + 5 = 0
(d) 3x + y + 3 = 0
Answer: a

Question: The locus of the point of intersection of the perpendicular tangents to the ellipse x²/9+ y²/4= 1 is
(a) x² + y² = 9
(b) x² + y² = 4
(c) x² + y² = 13
(d) x² + y² = 5
Answer: c

Question: The number of circle having radius 5 and passing through the points (– 2, 0) and (4, 0) is:
(a) One
(b) Two
(c) Four
(d) Infinite
Answer: b

Question: The equations of any tangents to the circle x² + y² − 2x + 4y − 4 = 0 is:
(a) y = m(x −1) + 3 √1+ m² − 2
(b) y = mx + 3 √1+ m²
(c) y = mx + 3 √1+ m² − 2
(d) None of these
Answer: a

Question: A circle of radius 5 units touches both the axes and lies in first quadrant. If the circle makes one complete roll on xaxis along the positive direction of x-axis, then its equation in the new position is:
(a) x² + y² + 20π x −10y +100π = 0
(b) x² + y² + 20π x +10y +100π = 0
(c) x² + y² − 20π x −10 y +100π = 0
(d) None of these
Answer: d

Question: The equation of the normal at the point (2, 3) on the ellipse 9x² + 16y² = 180 is
(a) 3y = 8x -10
(b) 3y - 8x + 7 = 0
(c) 8y + 3x + 7 = 0
(d) 3x + 2y + 7 = 0
Answer: b