Class 11 Mathematics Conic Sections MCQs Set D

Question: Let E be the ellipse x²/9 + y²/4 = 1 and C be the circle x² + y² = 9. Let P andQbe 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 Cand E
(d) P lies inside C but outside E
Answer: b

Question: The length of the axes of the conic 9x² + 4y² - 6x + 4y +1 = 0
(a) 1/2,9
(b) 3, 2/5
(c) 1, 2/3
(d) 3, 2
Answer: c

Question: The eccentric angles of the extremities of latusrectum of the ellipse x²/a² + y²/b² = 1 are given by
(a) tan-1 (± ac/b)
(b) tan-1 (± bc/a)
(c) tan-1 (± b/ae)
(d) tan-1 (± a/be)
Answer: c

Question: If the two tangents drawn to the ellipse x²/a² + y²/b² = 1 intersect perpendicularly at p, then locus of p is a circle x² + y² = a² + b². Then, the circle is called
(a) auxiliary circle
(b) director circle
(c) great circle
(d) None of these
Answer: b

Question: Consider the following statements
I. Circle x²+y²-x-y-1=0 is completely inside the circle x²+y²- x+2y-7=0.
II. Number of common tangents of the circles x²+y²+14x+12y+21=0 and x²+y²+2x-4y-4=0 is 4.
Which of these is/are correct?
(a) Only I
(b) Only II
(c) Both I and II
(d) None of these 
Answer: a

Question: The range of values of a such that the angle θ between the pair of tangents drawn from( a,0 ) to the circle x²+y²=1 satisfies π/2<θ π,is
(a) (1, 2)
(b) ( 1,√2)
(c) ( -√2,-1 ) 
(d) (-√2-1) ∪ (1,√2)
Answer: d

Question: Points (– 6, 0), (0, 6) and (–7, 7) are the vertices of ∆ABC. The incircle of the triangle has the equation
(a) x²+ y² -9x -9y + 36 =0
(b) x²+ y² +9x -9y + 36 =0
(c) x²+ y² +9x +9y - 36 =0
(d) x²+ y² +18x -18y +36 =0 
Answer: b

Question: Two rods of lengths a b and slide along the x-axis and y-axis respectively in such a manner that their ends are concyclic. The locus of the centre of the circle passing through the end points is
(a) 4(x²+y²) =a² +b²
(b) x²+y2 =a² +b²
(c) 4(x²-y²) =a² -b2
(d) x²-y² =a² -b² 
Answer: c

Question: A straight line with slope 2 and y-intercept 5 touches the circle, x²+y²+16x+12y+c=0 at a point Q.Then, the coordinates of Q are
(a) (-6,11) 
(b) (-9,-13) 
(c) ( -10,10-15)
(d) (-6,-7) 
Answer: d

Question: Let P be a variable point on the ellipse x2/25 + y2/16 = 1with foci at S and S¢. If A be the area of DPSS¢, then the maximum value of A is
(a) 24 sq units
(b) 12 sq units
(c) 36 sq units
(d) None of these
Answer: b

Question: The equations of the sides AB, BC and CA of a ∆ ABC are x +y = 1, 4x -y + 4 = 0 and 2x+ 3y= 6.
Circles are drawn on AB BC , and CA as diameters. The point of concurrence of the common chord is
(a) centroid of the triangle
(b) orthocentre
(c) circumcentre
(d) incentre
Answer: b

Question: If OA OB and are equal perpendicular chord of the circles x²+ y²-2x + 4y= 0, then equations of OA and OB are (where, O is origin)
(a) 3x + y =0 and 3x-y =0 
(b) 3x + y =0 and 3x-x =0 
(c) x + 3y =0 and y-3x =0 
(d) x + y = 0 and x-y= 0 
Answer: b

Question: Equation of chord of the circle x²+y²-3x-4y-4=0, which passes through the origin such that the origin divides it in the ratio 4 1: ,is
(a) x = 0
(b) 24x+y=0 
(c) 7x+24=0
(d) 7x-24y=0 
Answer: b

Question: An isosceles right angled triangle is inscribed in the circle x²+y²= r². If the coordinates of an end of the hypotenuse are ( a ,b), the coordinates of the vertex are
(a) (-a,-b)
(b) (b,- a)
(c) (b, a)
(d) (-b,- a) 
Answer: b

Question: In a ∆ABC, right angled at A, on the leg AC as diameter, a semi-circle is described. If a chord joins A with the point of intersection D of the hypotenuse and the semi-circle, then the length of AC equal to
(a) ABx AD/√AB²+ AD²
(b) ABx AD/AB+ AD
(c) √ABx AD 
(d) AB xAD/√AB²-AD² 
Answer: d

Question: A line meets the coordinate axes in A B and .A circle is circumscribed about the ∆OAB. The distances from the points A and Bof the side AB to the tangent at O are equal to m n and respectively. Then, the diameter of the circle is
(a) m (m+ n)
(b) n (m+n) 
(c) m- n
d) None of these 
Answer: d

Question: A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at A enclosing an angle of 60°. The area enclosed by these tangents and the arc of the circle is
(a) 2/√3 −π/6
(b) √3 -π/3
(c) π/3- √3/6
(d) √3 (1- π/6)
Answer: b

Question: The equation of a circle C₁ is x²+y²=4. The locus of the intersection of orthogonal tangents to the circle is the curve C₂ and the locus of the intersection of perpendicular tangents to the curve C₂ is the curve C₃ .Then,
(a) C₃is a circle
(b) the area enclosed by the curve C₃ is 8π
(c) C₂ and C₃ are circles with the same centre
(d) None of the above
Answer: a,c

Question: Let L₁ be a straight line passing through the origin and L₂ be the straight line x+ y = 1. If the intercepts made by the circle x2+y^2-x+3y=0 on L₁ and L₂ are equal, then L₁ can be represented by
(a) x+ y = 0
(b) x- y = 0
(c) 7x+y=0
(d) x- 7y=0 
Answer: b

Question: The locus of a point which moves such that the sum of its distances from two fixed points is always a constant, is
(a) a straight line
(b) a circle
(c) an ellipse
(d) a hyperbola
Answer: c

Question: If equation of ellipse is x²/4 + y²/25 = 1 , then coordinate of the foci, eccentricity and the length of the latusrectum are respectively
(a) (0,± √21), √21/5 , 7/5
(b) (0,± √21), √21/5 , 8/5
(c) (0,± √21), √21/7 , 8/5
(d) None of the above
Answer: b

Question: If p, q are the segments of a focal chord of an ellipse b²x² + a²y² = a²b², then
(a) a²( p+q ) = 2bpq
(b) b²( p+q ) = 2apq
(c) a( p+q ) = 2b²pq
(d) b( p+q ) = 2a²pq
Answer: b

Question: The position of the point(4, - 3)relative to an ellipse x2/28 + y2/20 = 1 is
(a) inside
(b) on the ellipse
(c) outside
(d) cannot say
Answer: c

Question: The distance of the centre of ellipse x² + 2y² - 2 = 0 to those tangents of the ellipse which are equallyinclined from both the axes, is
(a) 3/√2
(b) √(3/2)
(c) √2/3
(d) √3/2
Answer: d

Question: Tangents are drawn from any point on the circle x² + y² = a² to the circle x² + y² = b² . If the chord of contact touches the circle x² + y² = c² , a > b, then:
(a) a, b, c are in (a)P.
(b) a, b, c are in G.P.
(c) a, b, c are in H.P.
(d) a, c, b are in G.P.
Answer: b

Question: If two distinct chords, drawn from the point (p, q) on the circle x² + y² = px + qy (where p, q ≠ 0) are bisected by the x-axis, then:
(a) p² = q²
(b) p² = 8q²
(c) p² < 8q²
(d) p² > 8q²
Answer: d

Question: A rhombus is inscribed in the region common to the two circles x²+y²-4x-12=0 and x²+y²+4x -12=0 with two of its vertices on the line joining the centres of the circles. The area of the rhombus is
(a) 8 √3 sq units
(b) 4 √3 sq units
(c) 6 √3 sq units
(d) None of these
Answer: a

Question: Two points P Q and are taken on the line joining the points A( 0,0) and B ( ,3a,0) such that AP= PQ=QB.
Circles are drawn on AP, PQ, QB and as diameters.
The locus of the points, the sum of the squares of the tangents from which to the three circles is equal to b² , is
(a) x² +y²-3ax+2a²-b²=0 
(b) 3(x²+y²)-9ax+8a^2-b²=0
(c) x²+y²-5ax+6a²-b²=0 
(d) x²+y²-ax-b²=0 
Answer: b

Question: The set of values of c so that the equations y= |x|+c and x2+y2-8|x| -9 =0 have no solution, is
(a) (- ∞, -3) ∪ (3,∞)
(b) ( -3,3), )
(c) (- ∞,5√2) ∪ (5√2,∞ )
(d) (5,√2-4, ∞) 
Answer: d

Question: If the normal at point P on the ellipse x²/a² + y²/b² = 1 meets the axes in R and S respectively, then PR : RS is equal to
(a) a : b
(b) a² : b² 
(c) b² : a²
(d) b : a
Answer: c

Question: The curve represented by x = 3(cos t + sin t)and y = 4(cos t - sin t) is
(a) ellipse
(b) parabola
(c) hyperbola
(d) circle
Answer: a

Question: The equation of the ellipse whose focus is (1, –1), the directrix of line x - y - 3 = 0 and eccentricity 1/2 is
(a) 7x² + 2xy + 7y² - 10x + 10y + 7 = 0
(b) 7x² + 2xy + 7y² + 7 = 0
(c) 7x² + 2xy + 7y² - 10x + 10y - 7 = 0
(d) None of the above
Answer: a

Question: The length of the latusrectum of the ellipse 3x² + y² = 12 is
(a) 4
(b) 3
(c) 8
(d) 4/√3
Answer: d

Question: An equation of the normal to the ellipse x2/a2 + y²/b2 = 1at the positive end of the latusrectum is
(a) x + ey + e³a = 0
(b) x - ey - ae³ = 0
(c) x - ey + e³a = 0
(d) x + ey - e³a = 0
Answer: b

Question: The circle x² + y² = c² contains the ellipse x²/a² + y²/b² = 1, if
(a) c < a
(b) c < b
(c) c > a
(d) c > b
Answer: c

Question: If ax² + by² + 2hxy + 2gx + 2fy + c = 0 (abc + 2fgh - af² - bg² - ch² ≠ 0) represents an ellipse, if
(a) h² = ab 
(b) h² >ab 
(c) h²< ab 
(d) None of these
Answer: c

Question: The equation of a line inclined at an angle π/4 to the x-axis, such that the two circles x²+y²=4, x²+y²-10x-14y+65=0 intercept equal lengths on it, is
(a) 2x-2y-3=0
(b) 2x-2y+3=0 
(c) x-y+6=0 
(d) x -y -6=0
Answer: a

Question: If the tangent line to an ellipse x2/a2 + y²/b2 = 1 1cuts intercepts h and k from axes, then a2/h2 + b2/k2 is equal to
(a) 0
(b) 1
(c) -1
(d) 2
Answer: b

Question: The locus of mid-points of chords of the ellipse x²/a² + y²/b² = 1 that touch the circle x² + y² = b², is
(a) ( x²/a² + y²/b²) = x²/a⁴ + y²/b⁴
(b) ( x²/a² + y²/b²) =b²(x²/a⁴ + y²/b⁴)
(c) ( x²/a² + y²/b²) = b²(x²/a⁴ + y²/b⁴)
(d) None of these
Answer: b

Question: An ellipse is sliding along the coordinate axes. If the foci of the ellipse are (1, 1) and (3, 3), then area of the director circle of the ellipse is
(a) 2π sq units
(b) 4π sq units
(c) 6π sq units
(d) 8π sq units
Answer: d