VBQs Introduction to Three Dimensional Geometry Class 11 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 11 Mathematics with solutions. The following Introduction to Three Dimensional Geometry Class 11 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 11 examinations.

**Introduction to Three Dimensional Geometry VBQs Class 11 Mathematics**

**Question. The normals at three points P, Q and R of the parabola y ^{2} = 4ax meet at (h, k). The centroid of the ΔPQR lies on **

(a) x = 0

(b) y = 0

(c) x = –a

(d) y = a

**Answer**

B

**Question. To the lines ax ^{2} + 2hxy + by^{2} = 0, the lines a^{2}x^{2} + 2h(a + b)xy + b^{2}y^{2} = 0 are **

(a) equally inclined

(b) perpendicular

(c) bisector of the angle

(d) None of the above

**Answer**

A

**Question. If the vertices of a triangle are A(0, 4, 1), B(2, 3, –1) and C(4, 5, 0), then the orthocentre of ΔABC, is **

(a) (4, 5, 0)

(b) (2, 3, –1)

(c) (–2, 3, –1)

(d) (2, 0, 2)

**Answer**

B

**Question. If x ^{2}/a^{2} + y^{2}/b^{2} = 1 ( a >b ) and x^{2} – y^{2} = c^{2} cut at right angles, then **

(a) a

^{2}+ b

^{2}= 2c

^{2}

(b) b

^{2}– a

^{2}= 2c

^{2}

(c) a

^{2}– b

^{2}= 2c

^{2}

(d) a

^{2}b

^{2}= 2c

^{2}

**Answer**

C

**Question. The mid point of the chord 4x – 3y = 5 of the hyperbola 2x ^{2} – 3y^{2} = 12 is **

(a) (0,-5/3)

(b) (2, 1)

(c) (5/4 , 0)

(d) (11/4 , 2)

**Answer**

B

**Question. The angle of intersection of the circles x ^{2} + y^{2} – x + y – 8 = 0 and x^{2} + y^{2} + 2x + 2y – 11 = 0 is **

(a) tan

^{-1}(19/9)

(b) tan

^{–1}(19)

(c) tan

^{-1}(9/19)

(d) tan

^{–1}(9)

**Answer**

C

**Question. The number of normals drawn to the parabola y ^{2} = 4x from the point (1, 0) is **

(a) 0

(b) 1

(c) 2

(d) 3

**Answer**

B

**Question. The lines 2x – 3y – 5 = 0 and 3x – 4y = 7 are diameters of a circle of area 154 sq units, then the equation of the circle is **

(a) x^{2} + y^{2} + 2x – 2y – 62 = 0

(b) x^{2} + y^{2} + 2x – 2y – 47 = 0

(c) x^{2} + y^{2} – 2x + 2y – 47 = 0

(d) x^{2} + y^{2} – 2x + 2y – 62 = 0

**Answer**

C

**Question. If the circle x ^{2} + y^{2} = a^{2} intersects the hyperbola xy = c^{2} in four points (xi, yi),for i = 1, 2, 3 and 4, then y_{1} + y_{2} + y_{3} equals **

(a) 0

(b) c

(c) a

(d) c4

**Answer**

A

**Question. The pairs of straight lines x ^{2} – 3xy + 2y^{2} = 0 and x^{2} – 3xy + 2y^{2} + x – 2 = 0 form a **

(a) square but not rhombus

(b) rhombus

(c) parallelogram

(d) rectangle but not a square

**Answer**

C

**Question. If A(–2, 1), B(2, 3) and C(– 2, – 4) are three points. ****Then, the angle between BA and BC is**

(a) tan^{-1}(2/3)

(b) tan^{-1}(3/2)

(c) tan^{-1}(1/3)

(d) tan^{-1}(1/2)

**Answer**

A

**Question. The angle between lines joining the origin to the point of intersection of the line √3x + y = 2 and the curve y ^{2} – x^{2} = 4 is **

(a) tan

^{-1}2/√3

(b) π/6

(c) tan

^{-1}(√3/2)

(d) π/2

**Answer**

C

**Question. A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. What is its y-intercept ? **

(a) 1/2

(b) 2/3

(c) 1

(d) 4/3

**Answer**

D

**Question. The locus of mid-points of tangents intercepted between the axes of ellipse x ^{2}/a^{2} + y^{2}/b^{2} = 1 will be **

(a) a

^{2}/x

^{2}+ b

^{2}/y

^{2}= 1

(b) a

^{2}/x2 + b

^{2}/y

^{2}= 2

(c) a

^{2}/x

^{2}+ b

^{2}/y

^{2}= 3

(d) a

^{2}/x

^{2}+ b

^{2}/y

^{2}= 4

**Answer**

D

**Question. Equation of the chord of the hyperbola 25x ^{2} – 16y^{2} = 400 which is bisected at the point (6, 2) is **

(a) 6x – 7y = 418

(b) 75x – 16y = 418

(c) 25x – 4y = 400

(d) None of these

**Answer**

B

**Question. The AB, BC and CA of a DABC have respectively 3, 4 and 5 points lying on them. The number of triangles that can be constructed using these points as vertices is **

(a) 205

(b) 220

(c) 210

(d) None of these

**Answer**

A

**Question. The perimeter of the triangle with vertices at (1, 0, 0), (0, 1, 0) and (0, 0, 1) is **

(a) 3

(b) 2

(c) 2 √2

(d) 3 √2

**Answer**

D

**Question. The equation of the conic with focus at (1, –1), directrix along x – y + 1 = 0 and with eccentricity √2 , is **

(a) x^{2} – y2 = 1

(b) xy = 1

(c) 2xy – 4x + 4y + 1 = 0

(d) 2xy + 4x – 4y – 1 = 0

**Answer**

C

**Question. The triangle formed by the tangent to the curve f(x) = x ^{2} + bx – b at the point (1, 1) and the coordinate axes lies in the first quadrant. **

**If its area is 2, then the value of b is**

(a) 1

(b) 3

(c) –3

(d) –2

**Answer**

C

**Question. If (–3, 2) lies on the circle x ^{2} + y^{2} + 2gx + 2fy + c = 0, which is concentric with the circle x^{2} + y^{2} + 6x + 8y – 5 = 0, then c is equal to **

(a) 11

(b) –11

(c) 24

(d) 100

**Answer**

B

**Question. The equations of the circle which pass through the origin and makes intercepts of lengths 4 and 8 on the x and y-axes respectively are **

(a) x^{2} + y^{2} ± 4x ± 8y = 0

(b) x^{2} + y^{2} ± 2x ± 4y = 0

(c) x^{2} + y^{2} ± 8x ± 16y = 0

(d) x^{2} + y^{2} ± x ± y = 0

**Answer**

A

**Question. The area (in square unit) of the circle which touches the lines 4x + 3y = 15 and 4x + 3y = 5 is **

(a) 4π

(b) 3π

(c) 2π

(d) π

**Answer**

D

**Question. The tangent at (1, 7) to the curve x2 = y – 6 touches the circle x ^{2} + y^{2} + 16x + 12y + c = 0 at **

(a) (6, 7)

(b) (–6, 7)

(c) (6, –7)

(d) (–6, –7)

**Answer**

D

**Question. The area (in square unit) of the triangle formed by x + y + 1 = 0 and the pair of straight lines x ^{2} – 3xy + 2y^{2} = 0 is **

(a) 7/12

(b) 5/12

(c) 1/12

(d) 1/6

**Answer**

C

**Question. If 2x + y + k = 0 is a normal to the parabola y ^{2} = – 8x, then the value of k, is **

(a) 8

(b) 16

(c) 24

(d) 32

**Answer**

C

**Question. The equation of straight line through the intersection of the lines x – 2y = 1 and x + 3y = 2 and parallel to 3x + 4y = 0 is **

(a) 3x + 4y + 5 = 0

(b) 3x + 4y – 10 = 0

(c) 3x + 4y – 5 = 0

(d) 3x + 4y + 6 = 0

**Answer**

C

**Question. The equation of the circle which passes through the origin and cuts orthogonally each of the circles x ^{2} + y^{2} – 6x + 8 = 0 and x^{2} + y^{2} – 2x – 2y = 7 is **

(a) 3x

^{2}+ 3y

^{2}– 8x –13y = 0

(b) 3x

^{2}+ 3y

^{2}– 8x + 29y = 0

(c) 3x

^{2}+ 3y

^{2}+ 8x + 29y = 0

(d) 3x

^{2}+ 3y

^{2}– 8x – 29y = 0

**Answer**

B

**Question. If the line lx + my – n = 0 will be a normal to the hyperbola, then a ^{2}/l^{2} – b^{2}/m^{2} = (a^{2}+b^{2})2/k , where k is equal to **

(a) n

(b) n

^{2}

(c) n

^{3}

(d) None of these

**Answer**

B

**Question. The number of common tangents to the circles x ^{2} + y^{2} = 4 and x^{2} + y^{2} – 6x – 8y = 24 is **

(a) 0

(b) 1

(c) 3

(d) 4

**Answer**

B

**Question. The centres of a set of circles, each of radius 3, lie on the circle x ^{2} + y^{2} = 25. The locus of any point in the set is **

(a) 4 ≤ x

^{2}+ y

^{2}≤ 64

(b) x

^{2}+ y

^{2}≤ 25

(c) x

^{2}+ y

^{2}≥ 25

(d) 3 ≤ x

^{2}+ y

^{2}≤ 9

**Answer**

A

**Question. If PQ is a double ordinate of hyperbola x ^{2}/a^{2} – y^{2}/b^{2} = 1 such that OPQ is an equilateral triangle, O being the centre of the hyperbola, then the eccentricity ‘e’ of the hyperbola satisfies **

(a) 1< e <2/√3

(b) e =2/√3

(c) e = √3/2

(d) e > 2/√3

**Answer**

D

**Question. The equation of straight line through the intersection of line 2x + y = 1 and 3x + 2y = 5 passing through the origin is **

(a) 7x + 3y = 0

(b) 7x – y = 0

(c) 3x + 2y = 0

(d) x + y = 0

**Answer**

A

**Question. The length of the parabola y ^{2} = 12x cut off by the latus-rectum is **

(a) 6(√2 + log(1+ 2))

(b) 3(√2 + log(1+ 2))

(c) 6(√2 – log(1+ 2))

(d) 3(√2 – log(1+ 2))

**Answer**

A

**Question. If the equation of an ellipse is 3x ^{2} + 2y^{2} + 6x – 8y + 5 = 0, then which of the following are true? **

(a) e = 1/√3

(b) centre is (–1, 2)

(c) foci are (–1, 1) and (–1, 3)

(d) All of the above

**Answer**

D

**Question. The line joining two points A(2, 0), B(3, 1) is rotated about A in anti-clock wise direction through an angle of 15°. The equation of the line in the new position , is **

(a) √3x – y – 2√3 = 0

(b) x – 3 √y – 2 = 0

(c) √3x + y – 2√3 = 0

(d) x + √3y – 2 = 0

**Answer**

A

**Question. The equation of the common tangents to the two hyperbolas x ^{2}/a^{2} – y^{2}/b^{2} = 1 and y^{2}/a^{2} – x^{2}/b^{2} = 1 , are **

(a) y = ± x ± √b

^{2}– a

^{2}

(b) y = ± x ± √a

^{2}– b

^{2}

(c) y = ± x ± √a

^{2}+ b

^{2}

(d) y = ± x ± (a

^{2}– b

^{2})

**Answer**

B

**Question. The length of the straight line x – 3y = 1 intercepted by the hyperbola x ^{2} – 4y^{2} = 1 is **

(a) √10

(b) 6/5

(c) 1/√10

(d) 6/5 √10

**Answer**

D

**Question. The line joining (5, 0) to (10 cos q, 10 sin q) is divided internally in the ratio 2 : 3 at P. If q varies, then the locus of P is **

(a) a straight line

(b) a pair of straight lines

(c) a circle

(d) None of the above

**Answer**

C

**Question. A straight line through the point A(3, 4) is such that its intercept between the axes is bisected at A. Its equation is **

(a) 3x – 4y + 7 = 0

(b) 4x + 3y = 24

(c) 3x + 4y = 25

(d) x + y = 7

**Answer**

B

**Question. A rod of length l slides with its ends on two perpendicular lines, then the locus of its mid point is **

(a) x^{2} + y^{2} = l2/4

(b) x^{2} + y^{2} = l2/2

(c) x^{2} – y^{2} = l2/4

(d) None of these

**Answer**

A

**Question. The equation λx ^{2} + 4xy + y^{2} + λx + 3y + 2 = 0 represents a parabola, if λ is **

(a) 0

(b) 1

(c) 2

(d) 4

**Answer**

D

**Question. The line 2x + 6 y = 2 is a tangent to the curve x ^{2} – 2y^{2} = 4. The point of contact is **

(a) (4, – √6)

(b) (7, – 2 √6)

(c) (2, 3)

(d) (√6 ,1)

**Answer**

A

**Question. The eccentricity of the ellipse, which meets the straight line x/y + y/2 = 1 on the axis of x and the straight line x/3 – y/5 = 1 on the axis of y and whose axes lie along the axes of coordinates, is **

(a) 3√2/7

(b) 2√6/7

(c) √3/7

(d) None of the above

**Answer**

B

**Question. If the normal at (ap ^{2}, 2ap) on the parabola y^{2} = 4ax, meets the parabola again at (aq^{2}, 2aq), then **

(a) p

^{2}+ pq + 2 = 0

(b) p

^{2}– pq + 2 = 0

(c) q

^{2}+ pq + 2 = 0

(d) p

^{2}+ pq + 1 = 0

**Answer**

A

**Question. The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is **

(a) 133

(b) 190

(c) 233

(d) 105

**Answer**

B

**Question. The point (3, – 4) lies on both circles x ^{2} + y^{2} – 2x + 8y + 13 = 0 and x^{2} + y^{2} – 4x + 6y + 11 = 0 Then, the angle between the circles is **

(a) 60°

(b) tan-1(1/2)

(c) tan-1(3/5)

(d) 135°

**Answer**

D

**Question. If a tangent having slope of – 4/3 to the ellipse x ^{2}/18 + y^{2}/32 = 1 intersects the major and minor axes in points A and B respectively, then the area of ΔOAB is equal to (O is centre of the ellipse) **

(a) 12 sq units

(b) 48 sq units

(c) 64 sq units

(d) 24 sq units

**Answer**

D

**Question. If two circles 2x ^{2} + 2y^{2} – 3x + 6y + k = 0 and x^{2} + y^{2} – 4x + 10y + 16 = 0 cut orthogonally, then the value of k is **

(a) 41

(b) 14

(c) 4

(d) 1

**Answer**

C

**Question. The curve described parametrically by x = t ^{2} + 2t – 1, y = 3t + 5 represents **

(a) an ellipse

(b) a hyperbola

(c) a parabola

(d) a circle

**Answer**

C

**Question. The two curves y = 3x and y = 5x intersect at an angle **

(a) tan^{-1}(log3 – log5/1+log3 log5)

(b) tan^{-1}(log3 + log5/1- log3 log5)

(c) tan^{-1}(log3 + log5/1+log3 log5)

(d) tan^{-1}(log3 – log5/1- log3 log5)

**Answer**

A

**Question. The directrix of the parabola y ^{2} + 4x + 3 = 0 is **

(a) x – 4/3 = 0

(b) x + 1/4 = 0

(c) x – 3/4 = 0

(d) x – 1/4 = 0

**Answer**

D

**Question. The equation of a directrix of the ellipse x ^{2}/16 + y^{2}/25 = 1 is **

(a) 3y = 5

(b) y = 5

(c) 3y = 25

(d) y = 3

**Answer**

C

**Question. If the axes are shifted to the point (1, –2) without solution, then the equation 2x ^{2} + y^{2} – 4x + 4y = 0 becomes **

(a) 2X

^{2}+ 3Y

^{2}= 6

(b) 2X

^{2}+ Y

^{2}= 6

(c) X

^{2}+ 2Y

^{2}= 6

(d) None of these

**Answer**

B

**Question. Let A and B are two fixed points in a plane then locus of another point C on the same plane such that CA + CB = constant, (> AB) is **

(a) circle

(b) ellipse

(c) parabola

(d) hyperbola

**Answer**

B

**Question. The locus of the mid-points of the focal chord of the parabola y ^{2} = 4ax is **

(a) y

^{2}= a(x – a)

(b) y

^{2}= 2a(x – a)

(c) y

^{2}= 4a(x – a)

(d) None of these

**Answer**

B