Class 11 Mathematics Conic Sections MCQs Set 03

Question: The equation x² + y² + 2gx + 2fy + c = 0 will represent a real circle, if
a) g² + f² – c < 0
b) g² + f² – c ≥ 0
c) always
d) None of the options
Answer: b

Question: The equation of the tangents drawn from the origin to the circle x² + y²– 2rx – 2hy + h² = 0, are
a) x = 0
b) y = 0
c) (h² – r² )x – 2rhy = 0
d) (h² – r² )x + 2rhy = 0
Answer(a,c)

Question: The circle passing through (1, -2) and touching the axis to x at (3, 0) also passes through the point
a) (-5,2)
b) (2, -5)
c) (5, – 2)
d) (-2,5)
Answer: c

Question: In a ΔABC, 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) AB x AD /√(AB² x AD²)
b) AB x AD /AB + AD
c) √(AB x AD)
d) AB x AD /√(AB² – AD²)
Answer: d

Question: How many common tangents can be drawn to the following circles x² + y² = 6 and x² + y² + 6x + 2y + 1 = 0 ?
a) 4
b) 3
c) 2
d) 1
Answer: a

Question: If the abscissae and ordinates of two points P and Q are roots of the equations x² + 2ax – b² = 0 and y² + 2py – q² = 0 respectively, then the equation of the circle with PQ as diameter, is
a) x² + y² + 2ax + 2py – b² – q² = 0
b) x² + y² – 2ax – 2py + b² + q² = 0
c) x² + y² – 2ax – 2py – b² – q²= 0
d) x² + y² – 2ax + 2py + b² + q² = 0
Answer: a

Question: A variable circle passes through the fixed point A( p, q) and touches x-axis. The locus of the other end of the diameter through A is
a) (x – p)² = 4qy
b) (x – q)² = 4py
c) ( y – p)² = 4qx
d) ( y – q)² = px
Answer: a

Question: The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is
a) 10/3
b) 3/5
c) 6/5
d) 5/3
Answer: a

Question: If (a cos Θi , a sin Θi) , i = 1, 2, 3 represent the vertices of an equilateral triangle inscribed in a circle, then
a) cos Θ₁ cos Θ₂ cosΘ₃ = 0
b) sec Θ₁ sec Θ₂ secΘ₃ = 0
c) tan Θ₁ tan Θ₂ tanΘ₃ = 0
d) cot Θ₁ cot Θ₂ cotΘ₃ = 0
Answer: a

Question: The point diameterically opposite to the point P(1, 0) on the circle x² + y² +2x + 2y – 3 = 0 is
a) (3, 4)
b) (3, – 4)
c) (- 3, 4)
d) (- 3, – 4)
Answer: d

Question: The tangents to x² + y² = a² having inclinationsa andbintersect at P. If cot a + cot b = 0, then the locus of P is
a) x + y = 0
b) x – y = 0
c) xy = 0
d) None of the options
Answer: c

Question: If P = (2, 3), then the centre of circumcircle of ΔQRS is
a) (2/13 .7/26)
b) (2/13 .3/26)
c) (3/13 .9/26)
d) (3/13 .2/13)
Answer: c

Question: LetC be the circle with centre (0, 0) and radius 3. The equation of the locus of the mid-points of the chords of the circleC that subtend an angle 2p/3 at its centre is
a) x² + y² =27/4
b) x² + y² =9/4
c) x² + y² =3/2
d) x² + y² =1
Answer: b

Question: If the lines 3x – 4 y – 7 = 0 and 2x – 3 y – 5 = 0 are two diameters of a circle of area 49p sq units, the equation of the circle is
a) x² + y² =2x – 2y – 62 = 0
b) x² + y² =2x + 2y – 62 = 0
c) x² + y² =2x + 2y – 47 = 0
d) x² + y² =2x – 2y – 47 = 0
Answer: c

Question: If the pair of lines ax² + 2(a+b)xy + by² = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector, then
a) 3a² + 2ab + 3b² = 0
b) 3a² + 10ab + 3b² = 0
c) 3a² – 2ab + 3b² = 0
d) 3a² – 10ab + 3b² = 0
Answer: a

Question: A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is
a) a parabola
b) a hyperbola
c) a circle
d) an ellipse
Answer: a

Question: The intercept on the line y = x by the circle x² + y² – 2x = 0 is AB. Equation of the circle on ABas a diameter is
a) x² + y² – x – y = 0
b) x² + y² – x + y = 0
c) x² + y² + x + y = 0
d) x² + y² + x – y = 0
Answer: a

Question: The circle x² + y² – 10x – 14 y + 24 = 0 cuts an intercepts on y-axis of length
a) 5
b) 10
c) 1
d) None of the options
Answer: b

Question: Consider a family of circles which are passing through the point (- 1, 1) and are tangent to x-axis. If (h, k) is the centre of circle, then
a) k ≥ 1/2
b) – 1/2 ≤ k ≤ 1/2
c) k ≤ 1/2
d) 0 < k < 1/2
Answer: a

Question: The latusrectum of the parabola y²=5x+4y+1 is
a) 5/4
b) 10
c) 5
d) 5/2
Answer: a

Question: If a > 2 b > 0, then the positive value of m for which y = mx – b√(1 + m2) is a common tangent to x2 + y2 = b2 and (x – a)2 + y2 = b2, is

Answer: a

Question: A tangent to a parabola y2=4 ax is inclined at π/3 with the axis of the parabola. The point of contact is

Answer: a

Directions : Each of these questions contains two statements : Statement I (Assertion) and Statement II (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select one of the codes (a), (b), (c) and (d) given below.
a) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.
b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
c) Statement I is true; Statement II is false.
d) Statement I is false; Statement II is true.

Question:
Statement I Circles x² + y² = 4 and x² + y² – 8x + 7 = 0 intersect each other at two distinct points.
Statement II Circles with centresC₁ andC₂ and radii r₁ and r₂ intersect at two distinct points, if |C₁ C₂ |<r₁ r₂ .
Answer: c

Question: Let C_1C2 and be two circles with C₂ lying inside C₁.
Statement I A circle C lying inside C₁ touches C₁ internally and C₂ externally. Then, the locus of the centre of C is an ellipse.
Statement II If A and B are foci and P be any point on the ellipse, then AP + BP = constant.
Answer: a

Question:
Statement I Number of circles passing through (- 2, 1), (- 1, 0), (- 4, 3) is 1.
Statement II Through three non-collinear points in a plane only one circle can be drawn.
Answer: d

Question: Consider two circles
S ≡ x² + y² + 2gx + 2fy = 0 and S’ ≡ x² + y² + 2g’ x + 2f’y = 0
Statement I If two circles S and S¢ touch each other, then f < g = fg< .
Statement II Two circles touch each other, if line joining their centres is perpendicular to all possible common tangents.
Answer: b

Question:
Statement I Circle x² + y² – 6x – 4y + 9 = 0 bisects the circumference of the circle x² + y² – 8x – 6y + 23 = 0.
Statement II Centre of first circle lie on the second circle.
Answer: b

Question:
Statement I A ray of light incident at the point (–3, –1) gets reflected from the tangent at (0, –1) to the circle x² + y² = 1. If the reflected ray touches the circle,then equation of the reflected ray is 4 y – 3x = 5.
Statement II The angle of incidence = angle of reflection i.e., ∠i = ∠r
Answer: b

Question: Consider L₁ 2x + 3y + p – 3 and L₂ : 2x + 3y + p + 3 = 0, where pis a real number and C : x² + y² + 6x -10y + 30 = 0
Statement I If line L₁ is a chord of circle C, then L₂ is not always a diameter of circle C.
Statement II If line L₁ is a diameter of circleC, then L₂ is not a chord of circle C .
Answer: c