Conic Sections VBQs Class 11 Mathematics

VBQs Conic Sections 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 Conic Sections 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.

Conic Sections VBQs Class 11 Mathematics

Question. Let L be a normal to the parabola y² = 4x. If L passes through the point (9, 6), then L is given by?         
(a) y − x + 3 = 0
(b) y + 3x − 33 = 0
(c) y + x −15 = 0
(d) y − 2x +12 = 0

Question. If the circle x² + y² = a intersects the hyperbola xy = c² in four points P(x₁ , y₁),Q(x₂ , y₂),R(x₃ , y₃),S(x₄ , y₄), then?         
(a) x₁ + x₂ + x₃ + x₄ = 0
(b) y₁ + y₂ + y₃ + y₄ = 0
(c) x₁ x₂ x₃ x₄ = c
(d) y₁ y₂ y₃ y₄ = c

Question. An ellipse intersects the hyperbola 2x² – 2y² =1 orthogonally. The eccentricity of the ellipse is along the coordinate axes, then: 
(a) Equation of ellipse is x² + 2y² = 2
(b) The foci of ellipse are (± 1,0)
(c) Equation of ellipse is x² + 2y² = 4
(d) The foci of ellipse are (± √2,0

Question. Let the eccentricity of the hyperbola x²/a² + y²/b² = 1 be reciprocal to that of the ellipse x²  = 4y² = 4. If the hyperbola passes through a focus of the ellipse, then:
(a) (a) the equation of the hyperbola is x²/3 – y²/2 = 1
(b) a focus of the hyperbola is (2, 0)
(c) the eccentricity of the hyperbola is √5/3
(d) the equation of the hyperbola is x² – 3y² = 3

Question. Tangents are drawn to the hyperbola x²/9 – y²/4 = 1 parallel to the straight line 2x – y = 1. The points of contacts of the tangents on the hyperbola are: 

Question. The points, where the normals to the ellipse x² + 3y² = 37 be parallel to the line 6x – 5y + 7 = 0 is:       
(a) (5, 2)
(b) (2, 5)
(c) (1, 3)
(d) (–5, –2)

Question. If the circle x² + y² = a² intersects the hyperbola xy = c2 in four points P(x₁ , y₁ ),Q(x₂ , y₂ ), R(x₃ , y₃), s(x₄ , y₄) then: 
(a) x₁ + x₂ + x₃ + x₄ = 0
(b) y₁ + y₂ + y₃ + y₄ = 0
(c) x₁, x₂,x_3,x₄ = c⁴
(d) y₁, y₂,y₃,y₄ = c⁴

Question. If the tangent at the point (4cosθ , 16/√11 sinθ) to the ellipse 16x² + 11y² = 256 is also a tangent to the circle x² + y² –256 is also a tangent to the circle x² + y² − 2x =15, then θ equals:   
(a) π/3
(b) 2π/3
(c) –π/3
(d) 5π/3

Question. The tangent PT and the normal PN to the parabola y² = 4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose:   
(a) vertex is (2a/3 , 0)
(b) directrix is x = 0
(c) latus rectum is 2a/3
(d) focus is (a, 0)

Question. Let A and B be two distinct points on the parabola y² = 4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be: 
(a)− 1/r
(b) 1/r
(c) 2/r
(d) 2/r

Question. The product of eccentricities of two conics is unity, one of them can be a/an?
(a) parabola
(b) ellipse
(c) hyperbola
(d) circle

Assertion and Reason

Note: Read the Assertion (A) and Reason (R) carefully to mark the correct option out of the options given below:
a. If both assertion and reason are true and the reason is the correct explanation of the assertion.
b. If both assertion and reason are true but reason is not the correct explanation of the assertion.
c. If assertion is true but reason is false.
d. If the assertion and reason both are false.
e. If assertion is false but reason is true.

Question. Assertion: The tangent to the parabola y² = 4x at any point P and perpendicular on it form the focus S meet on the directrix of the parabol(a)   
Reason: Tangents and normals at the extremities of the latus rectum of a parabola y2 = 4ax constitute a square whose area is 8a2sq. units

Question. Consider the two curves C₁ : y² = 4x,C₂ : x² + y² −6x+1=0       
Assertion: C₁ and C₂ touch each other exactly at two points.
Reason: Equation of the tangent at (1,2) to C₁ and C₂ both is x – y + 1 = 0 and at (1,–2) is x + y + 1 = 0

Question. Assertion: If the normal at an end of a latusrectum of the ellipse x²/a²+ y²/b²=1 meets the major axis at G,O is the centre of the ellipse, then OG=ae³,e being the eccentricity of the ellipse     
Reason: Equation of the normal at a point (acosθ , bsinθ ) on the ellipse x²/a²+ y²/b²=1 is ax/cosθ + by/sinθ = a²+b².

Question. Assertion: The curve y = x²/2 + x + 1 is symmetrical with respect to the line x = 1.   
Reason: A parabola is symmetric about its axis          

Question. Assertion: If the length of the latus rectum of an ellipse is 1/3 of the major axis, then the eccentricity of the ellipse is √2 / 3.    
Reason: If a focus of an ellipse is at the origin directrix is the line x = 4 and the eccentricity is √2 / 3, then the length of the semi major axis is 4. √6

Question. Assertion: A equation of a common tangent to the parabola y² =16√3x and the ellipse 2x² + y² = 4 is y = 2x + 2√3. 
Reason: If the line y= mx + 4√3/m , (m≠0) is a common tangent to the parabola y² =16√3x and the ellipse 2x² + y2 = 4, then m satisfies m⁴ + 2m² = 24

Question. Assertion: Two tangents drawn from any point on the hyperbola x² − y² = a² − b² to the ellipse x²/a²-y²/b²=1 complementary angles with the axis of the ellipse       
Reason: If two lines make complementary angles with the axis of x then the product of their slopes is 1.

Question. Assertion: If the vertex of a parabola lies at the point (a, 0) and the directrix is y-axis then the focus of the parabola is at the point (2a, 0).       
Reason: Length of the common chord of the parabola y² = 12x and the circle x² + y² = 9 is equal to the length of  the latus rectum of the parabol(a)

Question. Assertion: If the foci an hyperbola are at the points (4, 1) and (–6,1), eccentricity is 5/4 then the length of the transverse axis is 4.   
Reason: Distance between the foci of a hyperbola is equal to the product of its eccentricity and the length of the transverse axis.

Question. Assertion: A parabola has the origin as its focus and the line y = 2 as the directrix, then the vertex of the parabola is at the point (0,1)       
Reason: Vertex of a parabola is equidistance form the focus and the directrix and lies on the line through thefoucs perpendicular to the directrix. 

Paragraph–I
Let PQ be a focal chord of the parabola y² = 4ax. The tangents to the parabola at P and Q meet at a point lying on the line y = 2x + a, a > 0.

Question. Length of chord PQ is:
(a) 7a
(b) 5a
(c) 2a
(d) 3a

Question. If chord PQ subtends an angle θ at the vertex of y² = 4ax, then tan θ is equal to:
(a) 2/3√7
(b)−2/3√7
(c) 2/3√5
(d)−2/3√5

Paragraph–II
Tangents are drawn from the point P(3,4) to the ellipse  x²/9+ y²/4 = 1 touching the ellipse at points A and (b)

Question. The coordinates of A and B are:      

Question. The orthocentre of the triangle PAB is:
(a) (5 , 8/7)
(b) (7/5 , 25/8)
(c) (11/5 , 8/5)
(d) (8/25 , 7/5)

Question. If the circles x² + y² – 16x – 20y + 164 = r² and (x – 4)² + (y – 7)² = 36 intersect at two distinct points, then:
(a) r > 11
(b) 0 < r < 1
(c) r = 11
(d) 1 < r < 11

Question. If a circle C, whose radius is 3, touches externally the circle, x² + y² + 2x – 4y – 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle c, on the x-axis is equal to
(a) 5
(b) 2√3
(c) 3√2
(d) 2√5

Question. A circle passes through (–2, 4) and touches the y-axis at (0, 2). Which one of the following equations can represent a diameter of this circle ?
(a) 2x – 3y + 10 = 0
(b) 3x + 4y – 3 = 0
(c) 4x + 5y – 6 = 0
(d) 5x + 2y + 4 = 0

Question. Locus of the image of the point (2, 3) in the line (2x – 3y + 4) + k (x – 2y + 3) = 0, k ∈ R, is a :
(a) circle of radius √2.
(b) circle of radius √3.
(c) straight line parallel to x-axis
(d) straight line parallel to y-axis

Question. A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, y – 4x + 3 = 0, then its radius is equal to
(a) √5
(b) 1
(c) √2
(d) 2

Question. The common tangent to the circles x² + y² = 4 and x² + y² + 6x + 8y – 24 = 0 also passes through the point:
(a) (4, –2)
(b) (– 6, 4)
(c) (6, –2)
(d) (– 4, 6)

Question. The sum of the squares of the lengths of the chords intercepted on the circle, x² + y² = 16, by the lines, x + y = n, n ∈N, where N is the set of all natural numbers, is :
(a) 320
(b) 105
(c) 160
(d) 210

Question. The radius of a circle, having minimum area, which touches the curve y = 4 – x² and the lines, y = |x| is :
(a) 4( √2 +1)
(b) 2( √2 +1)
(c) 2( √2 -1)
(d) 4( √2 -1)

Question. The equation of circle described on the chord 3x + y + 5 = 0 of the circle x² + y² = 16 as diameter is:
(a) x² + y² + 3x + y – 11 = 0
(b) x² + y² + 3x + y + 1 = 0
(c) x² + y² + 3x + y – 2 = 0
(d) x² + y² + 3x + y – 22 = 0

Question. For the two circles x² + y² = 16 and x² + y² – 2y = 0, there is/are
(a) one pair of common tangents
(b) two pair of common tangents
(c) three pair of common tangents
(d) no common tangent

Question. The two ad acent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60° . If the area of the quadrilateral is 4√3 , then the perimeter of the quadrilateral is :
(a) 12.5
(b) 13.2
(c) 12
(d) 13

Question. Two parabolas with a common vertex and with axes along x-axis and y-axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is?
(a) 3 (x + y) + 4 = 0
(b) 8 (2x + y) + 3 = 0
(c) 4 (x + y) + 3 = 0
(d) x + 2y + 3 = 0

Question. The tangent to the circle C₁ : x² + y² – 2x – 1 = 0 at the point (2, 1) cuts off a chord of length 4 from a circle C₂ whose centre is (3, – 2). The radius of C₂ is
(a) √6
(b) 2
(c) √2
(d) 3

Question. If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centre and subtend angles cos−1(1/7) and sec−1(7) at the centre respectively, then the distance between these chords, is :
(a) 4/√7
(b) 8/√7
(c) 8/7
(d) 16/7

Question. Let PQ be a diameter of the circle x² + y² = 9. If α and β are the lengths of the perpendiculars from P and Q on the straight line, x + y = 2 respectively, then the maximum value of αβ is ______________.

Question. The diameter of the circle, whose centre lies on the line x + y = 2 in the first quadrant and which touches both the lines x = 3 and y = 2, is __________.

Question. The number of common tangents to the circles x² + y² – 4x – 6x – 12 = 0 and x² + y² + 6x + 18y + 26 = 0, is:
(a) 3
(b) 4
(c) 1
(d) 2

Question. If the incentre of an equilateral triangle is (1, 1) and the equation of its one side is 3x + 4y + 3 = 0, then the equation of the circumcircle of this triangle is :
(a) x² + y² – 2x – 2y – 14 = 0
(b) x² + y² – 2x – 2y – 2 = 0
(c) x² + y² – 2x – 2y + 2 = 0
(d) x² + y² – 2x – 2y – 7 = 0

Question. If y + 3x = 0 is the equation of a chord of the circle, x² + y² – 30x = 0, then the equation of the circle with this chord as diameter is :
(a) x² + y² + 3x + 9y = 0
(b) x² + y² + 3x – 9y = 0
(c) x² + y² – 3x – 9y = 0
(d) x² + y² – 3x + 9y = 0

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

Question. If the two circles (x -1)² + ( y – 3)² = r² and x² + y² – 8x + 2y + 8 = 0 intersect in two distinct point, then
(a) r > 2
(b) 2 < r < 8
(c) r < 2
(d) r = 2

Question. The lines 2x – 3y = 5 and 3x – 4y = 7 are diameters of a circle having area as 154 sq.units. Then the equation of the circle is
(a) x² + y² – 2x + 2y = 62
(b) x² + y² + 2x – 2y = 62
(c) x² + y² + 2x – 2y = 47
(d) x² + y² – 2x + 2y = 47

Question. The number of common tangents of the circles given by x² + y² – 8x – 2y + 1 = 0 and x² + y² + 6x + 8y = 0 is
(a) one
(b) four
(c) two
(d) three

Question. The two circles x² + y² = ax and x² + y² = c² (c > 0) touch each other if
(a) | a | = c
(b) a = 2c
(c) | a | = 2c
(d) 2 | a | = c

Question. Let L₁ be a tangent to the parabola y² = 4(x + 1) and L₂ be a tangent to the parabola y² = 8(x + 2) such that L₁ and L₂ intersect at right angles. Then L₁ and L₂ meet on the straight line :
(a) x + 3 = 0
(b) 2x + 1 = 0
(c) x + 2 = 0
(d) x + 2y = 0

Question. The centre of the circle passing through the point (0, 1) and touching the parabola y = x² at the point (2,4) is:
(a) (-53/10, 16/5)
(b) (6/5, 53/10)
(c) (3/10, 16/5)
(d) (-16/5, 53/10)

Question. If the lines 2x + 3y +1 = 0 and 3x – y – 4 = 0 lie along diameter of a circle of circumference 10π, then the equation of the circle is
(a) x² + y² + 2x – 2y – 23 = 0
(b) x² + y² – 2x – 2y – 23 = 0
(c) x² + y² + 2x + 2y – 23 = 0
(d) x² + y² – 2x + 2y – 23 = 0

Question. Intercept on the line y = x by the circle x² + y² – 2x = 0 is AB. Equation of the circle on AB as 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

Question. The circle x² + y² = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points if
(a) – 35 < m < 15
(b) 15 < m < 65
(c) 35 < m < 85
(d) – 85 < m < – 35

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) an ellipse
(b) a circle
(c) a hyperbola
(d) a parabola

Question. If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas y² = 4ax and x² = 4ay, then
(a) d² + (3b – 2c)² = 0
(b) d² + (3b + 2c)² = 0
(c) d² + (2b – 3c)² = 0
(d) d² + (2b + 3c)² = 0

Question. If P and Q are the points of intersection of the circles x² + y² + 3x + 7y + 2p – 5 = 0 and x² + y² + 2x + 2y – p²=0 then there is a circle passing through P, Q and (1, 1) for:
(a) all except one value of p
(b) all except two values of p
(c) exactly one value of p
(d) all values of p

Question. Three distinct points A, B and C are given in the 2-dimensional coordinates plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point (–1, 0) is equal to 1/3. Then the circumcentre of the triangle ABC is at the point:
(a) (5/4, 0)
(b) (5/2, 0)
(c) (5/3, 0)
(d) (0, 0)

Question. The equation of the circle passing through the point (1, 2) and through the points of intersection of x² + y² – 4x – 6y – 21 = 0 and 3x + 4y + 5 = 0 is given by
(a) x² + y² + 2x + 2y + 11 = 0
(b) x² + y² – 2x + 2y – 7 = 0
(c) x² + y² + 2x – 2y – 3 = 0
(d) x² + y² + 2x + 2y – 11 = 0

Question. The equation of the circle passing through the point (1, 0) and (0, 1) and having the smallest radius is –
(a) x² + y² – 2x – 2y +1 = 0
(b) x² + y² – x – y = 0
(c) x² + y² + 2x + 2y – 7= 0
(d) x² + y² + x + y – 2 = 0

Question. Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of 2π/3 at its center is
(a) x² + y² = 3/2
(b) x² + y² = 1
(c) x² + y² = 27/4
(d) x² + y² = 9/4

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

Question. The set of all real values of λ for which exactly two common tangents can be drawn to the circles x² + y² – 4x – 4y + 6 = 0 and x² + y² – 10x – 10y + λ = 0 is the interval:
(a) (12, 32)
(b) (18, 42)
(c) (12, 24)
(d) (18, 48)

Question. If the point (1, 4) lies inside the circle x² + y² – 6x – 10y + P = 0 and the circle does not touch or intersect the coordinate axes, then the set of all possible values of P is the interval:
(a) (0, 25)
(b) (25, 39)
(c) (9, 25)
(d) (25, 29)

Question. Let the tangents drawn from the origin to the circle, x² + y² – 8x – 4y + 16 = 0 touch it at the points A and B. The (AB)² is equal to:
(a) 52/5
(b) 56/5
(c) 64/5
(d) 32/5

Question. If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90^o, then the length (in cm) of their common chord is :
(a) 13/5
(b) 120/13
(c) 60/13
(d) 13/2

Question. The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :
(a) √5/2
(b) 2√5
(c) √5/4
(d) 4√5

Question. Let z ∈ C, the set of complex numbers. Then the equation, 2|z+3i| – |z– i| = 0 represents :
(a) a circle with radius 8/3.
(b) a circle with diameter 10/3.
(c) an ellipse with length of major axis 16/3.
(d) an ellipse with length of minor axis 16/9

Question. If a point P has co–ordinates (0, –2) and Q is any point on the circle, x² + y² – 5x – y + 5 = 0, then the maximum value of (PQ)² is :
(a) (25+√6) / 2
(b) 14 + 5√3
(c) (47+10√6) / 2
(d) 8 + 5√3

Question. If the tangent at (1, 7) to the curve x² = y – 6 touches the circle x² + y² +16x +12y + c = 0 then the value of c is :
(a) 185
(b) 85
(c) 95
(d) 195

Question. The locus of the centres of the circles, which touch the circle, x² + y² = 1 externally, also touch the y-axis and lie in the first quadrant, is:
(a) x = √(1+ 4y), y ≥ 0
(b) y = √(1+ 2x), x ≥ 0
(c) y = √(1+ 4x), x ≥ 0
(d) x = √(1+ 2y), y ≥ 0

Question. All the points in the set S = { (a+i)/(a−i) : ∝ ∈ R}(i = √−1) lie on a:
(a) straight line whose slope is 1.
(b) circle whose radius is 1.
(c) circle whose radius is √2 .
(d) straight line whose slope is –1.

Question. The equation of the locus of the point whose distance from the point P and the line AB are equal, is:
(a) 9x² + y² − 6xy − 54x − 62y + 241 = 0
(b) x² + 9y² + 6xy − 54x + 62y − 241 = 0
(c) 9x² + 9y² − 6xy − 54x − 62y − 241 = 0
(d) x² + y² − 2xy + 27x + 31y −120 = 0

Paragraph–III

The circle x² + y² – 8x = 0 and hyperbola x²/9 – y²/4 = 1 intersect at the points A and (b)

Question. Equation of a common tangent with positive slope to the circle as well as to the hyperbola is:
(a) 2x −√5y − 20 = 0
(b) 2x − √5y + 4 = 0
(c) 3x – 4y + 8 = 0
(d) 4x – 3y + 4 = 0

Question. Equation of the circle with AB as its diameter is:
(a) x² + y² – 12x + 24 = 0
(b) x² – y² + 12x + 24 = 0
(c) x² + y² + 24x – 12 = 0
(d) x² + y² – 24x – 12 = 0

Paragraph–IV
The difference between the second degree curve and pair of asymptotes is constant. If second degree curve represented by a hyperbola S = 0, then the equation of its asymptotes is S + λ = 0, and if equation of conjugate which will be a pair of straight lines, then we get λ. Then equation of asymptotes is A ≡ S + λ = 0 and if equation of conjugate hyperbola of S represented by S1, then A is the arithmetic means of S and S1.

Question. Pair of asymptotes of the hyperbola xy – 3y – 2x = 0 is:
(a) xy – 3y – 2x + 2 = 0
(b) xy – 3y – 2x + 4 = 0
(c) xy – 3y – 2x + 16 = 0
(d) xy – 3y – 2x + 12 = 0

Question. The asymptotes of a hyperbola having centre at the point (1, 2) are parallel to the lines 2x + 3y = 0 and 3x + 2y = 0.
If the hyperbola passes through the point (5,3) then its equation is:
(a) (2x + 3y – 3) (3x + 2y – 5) = 256
(b) (2x + 3y – 7) (3x + 2y – 8) = 156
(c) (2x + 3y – 5) (3x + 2y – 3) = 256
(d) (2x + 3y – 8) (3x + 2y – 7) = 154

Question. If angle between the asymptotes of hyperbola x²/a²+ y²/b²=1 is π/3 thant he centricity of conjugate hyperbola is:
(a) √2
(b) 2
(c) 2/√3
(d) 4/√3

Question. A hyperbola passing through origin has 3x – 4y – 1 = 0 and 4x – 3y – 6 = 0 as its asymptotes. Then the equation of its tansverse and conjugate axes as:
(a) x – y – 5 = 0 and x + y + 1 = 0
(b) x – y = 0 and x + y + 5 = 0
(c) x + y – 5 = 0 and x – y – 1 = 0
(d) x + y – 1 = 0 and x – y – 5 = 0

Question. The tangent at any point of a hyperbola 16x² – 25y² = 400 cuts off a triangle form the asymptotes and that the portion of it intercepted between the asymptotes, then the area of this triangle is:
(a) 10 aq. unit
(b) 20 sq. unit
(c) 30 sq. unit
(d) 40 sq unit

Match the Column

Question. Match the statement of Column with those in Column II:

(a) A→ 2,3,4; B→ 1,2; C→ 4,5
(b) A→ 1,3; B→ 2,4,5; C→ 2,3
(c) A→ 3,4,5; B→ 1,3; C→ 2,3
(d) A→ 1,5; B→ 2,4,5; C→3,4

Question. Normals at P, Q, R are drawn to 2 y = 4x which intersect at (3, 0). Then: 

(a) A→ 1; B→ 2; C→ 4; D→ 3
(b) A→ 2; B→ 4; C→ 3; D→ 1
(c) A→ 3; B→ 4; C→ 2; D→ 1
(d) A→ 4; B→ 1; C→ 3; D→ 2

Question. Match the statement of Column with those in Column II:

(a) A→ 1,3; B→ 5; C→ 2,4
(b) A→ 2,3; B→ 4,5; C→ 3
(c) A→ 1,2; B→ 3; C→ 2,4
(d) A→ 4,5; B→ 1,3; C→ 3

Question. The normal to the parabola y2 = 8x at the point (2, 4) meets it again at (18, –12). If length of normal chord is λ , then the value of λ² must be:         

Question. Let Δ1 be the are of ΔPQR inscribed in an ellipse and Δ₂ be the area of the ΔP’Q’R’whose vertices are the points lying on the auxiliary circle corresponding to the points P, Q R, respectively. If the eccentricity of the ellipse is 4√3/7 then the ratio 343 Δ₂/Δ₁ must be:     

Question. From a point A common tangents are drawn to the circle x² + y² = a²/2 and the parabola y² = 4ax. If the area of the quadrilateral formed by the common tangents, the chords of contact of the point A, w.r.t. the circle and the parabola is λ square unit, then the value of 256/a²λ must be:   

Question. Three normals with slopes 1 2 m , m and 3 m are drawn form a point P not on the axis of parabola y² = 4x.    
If m₁m₂ =α ,results in the locus of P being a part of parabola, then the value of (36)^α must be:

Question. If the normals at the four points (x₁, y₁),(x₂, y₂),(x₃, y₃) and (x₄,y₄) on the ellipse x²/a²+ y²/b²=1 are concurrent, then the value of (x₁ + x₂ + x₃ + x₄ ) x (1/x₁ + 1/x₂ + 1/x₃ + 1/x₄) must be: 

Question. TP and TQ are any two tangents to a parabola and the tangent at a third point R cuts then in P′ and Q′, then the value of TP’/TP = TQ’/TQ must be:     

Question. If a circle cuts a rectangular hyperbola xy = c² in A, B, C and D are the parameters of these four points be t₁ , t₂, t₃ and t₄ respectively, then the value of
16t₁t₂t₃t₄ must be:

Question. A water jet from a fountain reaches its maximum height of 4 m at a distance 0.5 m from the vertical passing through the point O of water outlet. If λm be the height of the jet above the horizontal OX at a distance of 0.75 m from the point O, then the value of λ⁶ must be:   

Question. If the normal at an end of a latus rectum of an ellipse x²/a²+ y²/b²=1 passes through one extremity of the minor axis. If e be the eccentricity of the ellipse then the value of
 625(2e² +1)² must be:     

Question. Tangents are drawn to the ellipse x²/9+ y²/5=1 at ends of latusrectum. If the area of an quadrilateral by λ sq unit, then the value of λ must be:   

Question. If four Points be taken on a rectangular hyperbola such that the chord joining any two is perpendicular to the  chord joining the other two and if α, β, γ, δ be the inclinations to either asymptote of the straight line joining these points to the centre, then the value of tan α tan β × tan γ tan δ must be:   

Question. If e be the eccentricity of the ellipse 4(x − 2y +1)² + 9 (2x + y + 2)² = 25, then the value of 2187e² must be:   

Question. A triangle is inscribed in xy = c² and two to its sides are parallel to y = m₁x and y = m₂x. If m₁, m₂ are two values of x²–6x+1=0 and if third side envelopes the hyperbola xy = c²λ , then the value of 16λ² must be: 

Question. The equation of the hyperbola whose asymptotes are x + 2y + 3 = 0 and 3x+4y+5=0 which passes through the point (1,–1) is 3x² +10xy + 8y +14x + 22y +λ = 0, then the equation of the conjugate hyperbola is 3x² +10xy +8y² +μ x + 22y + v = 0, then the value of μ + v, must be: