CBSE Class 11 Mathematics HOTs Straight Lines

Question. Through the point P(α,β),where αβ > 0 , the straight line x/a + y/b = 1 is drawn so as to form with coordinate axes a triangle of area S. If ab > 0, then least value of S is
(a) 2αβ
(b) (1/2)αβ
(c) ab
(d) None of these
Answer : A

Question. If the straight lines 2x + 3y – 1 = 0, x + 2y – 1 = 0 and ax + by – 1 = 0 form a triangle with origin as orthocentre, then (a, b) is given by
(a) (6, 4)
(b) (– 3, 3)
(c) (– 8, 8)
(d) (0, 7)
Answer : C

Question. The vertices of a triangle are (ab, 1/ab), (bc, 1/bc) and (ca,1/ca) where a, b, c are the roots of the equation x³ – 3x² + 6x + 1 = 0. The coordinates of its centroid are.
(a) (1, 2)
(b) (2, – 1)
(c) (1, – 1)
(d) (2, 3)
Answer : B

Question. Consider points A (3, 4) and B (7, 13). If P be a point on the line y = x such that PA + PB is minimum, then coordinates of P are
(a) (12/7, 12/7)
(b) (13/7, 13/7)
(c) (31/7, 31/7)
(d) (0, 0)
Answer : C

Question. Let A (–3, 2) and B (–2, 1) be the vertices of a triangle ABC. If the centroid of this triangle lies on the line 3x + 4y + 2 = 0, then the vertex C lies on the line :
(a) 4x + 3y + 5 = 0
(b) 3x + 4y + 3 = 0
(c) 4x + 3y + 3 = 0
(d) 3x + 4y + 5 = 0
Answer : B

Question. The straight line y = x – 2 rotates about a point where it cuts the x-axis and becomes perpendicular to the straight line ax + by + c = 0. Then its equation is
(a) ax + by + 2a = 0
(b) ax – by – 2a = 0
(c) bx + ay – 2b = 0
(d) ay – bx + 2b = 0
Answer : D

Question. The range of values of β such that (0, β) lie on or inside the triangle formed by the lines y + 3x + 2 = 0, 3y – 2x – 5 = 0, 4y + x – 14 = 0 is
(a) 5 < b ≤ 7
(b) 1/1 ≤ β ≤ 1
(c) 5/3 ≤ β ≤ 7/2
(d) None of these
Answer : C

Question. The line x/a + y/b = 1 meets the axis of x and y at A and B respectively and the line y = x at C so that area of the trinagle AOC is twice the area of the triangle BOC, O being the origin, then one of the positions of C is
(a) (a, a)
(b) (2a/3, 2a/3) 
(c) (b/3, b/3) 
(d) (2b/3, 2b/3) 
Answer : D

Question. If three distinct points A, B, C are given in the 2-dimensional coordinate plane such that the ratio of the distance of each one of them from the point (1, 0) to the distance from (– 1, 0) is equal to 1/2, then the circumcentre of the triangle ABC is at the point
(a) (5/3 ,0)
(b) (0, 0)
(c) (1/3 ,0)
(d) (3, 0)
Answer : A

Question. The line parallel to the x- axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx – 2ay – 3a = 0, where (a, b) ¹ (0, 0) is
(a) below the x - axis at a distance of 3/2 from it
(b) below the x - axis at a distance of 2/3 from it
(c) above the x - axis at a distance of 3/2 from it
(d) above the x - axis at a distance of 2/3 from it
Answer : A

Question. The equation of bisector of that angle between the lines x + y + 1 = 0 and 2x – 3y – 5 = 0 which contains the point (10, – 20) is
(a) x (√13 + 2√2) + y (√13 – 3√2) + (√13 – 5√2) = 0
(b) x (√13 – 2√2) + y (√13 + 3√2) + (√13 + 5√2) = 0
(c) x (√13 + 2√2) + y (√13 + 3√2) + (√13 + 5√2) = 0
(d) None of these
Answer : A

Question. The intercepts on the straight line y = mx by the lines y = 2 and y = 6 is less than 5, then m belongs to
(a) (- 4/3, 4/3)
(b) (4/3, 4/3)
(c) (- ∞, 4/3) ∪ (4/3, ∞)
(d) (- 4/3, ∞)
Answer : C

Question. The base of an equilateral triangle is along the line given by 3x + 4y = 9. If a vertex of the triangle is (1, 2), then the length of a side of the triangle is:
(a) 2√3/15
(b) 4√3/15
(c) 4√3/5
(d) 2√3/5
Answer : B

Question. The circumcentre of a triangle lies at the origin and its centroid is the mid point of the line segment joining the points (a² + 1, a² + 1) and (2a, – 2a), a ¹ 0. Then for any a, the orthocentre of this triangle lies on the line:
(a) y – 2ax = 0
(b) y – (a² + 1)x = 0
(c) y + x = 0
(d) (a – 1)²x – (a + 1)²y = 0
Answer : D

Question. If the point (a, 2) lies between the lines x – y – 1 = 0 and 2 (x – y) + 5 = 0, then the set of values of ‘a’ is
(a) (– ∞ , 3) ∪ ( 9/2, ∞)
(b) (3, 9/2)
(c) (– ∞ , 3)
(d) (- 1/2 , 3)
Answer : D

Question. If two vertices of a triangle are (5, –1) and (–2, 3) and its orthocentre is at (0, 0), then the third vertex is
(a) (4, – 7)
(b) (– 4, – 7)
(c) (– 4, 7)
(d) (4, 7)
Answer : B

Question. The bisector of the acute angle formed between the lines 4x - 3y + 7 = 0 and 3x - 4y +14 = 0 has the equation :
(a) x + y + 3 = 0
(b) x - y -3 = 0
(c) x - y + 3 = 0
(d) 3x + y - 7 = 0
Answer : C

Numeric Value Answer

Question. The straight line L ≡ x + y + 1 = 0 and L₁ ≡ x + 2y + 3 = 0 are intersecting. m is the slope of the straight line L₂ such that L is the bisector of the anlge between L₁ and L₂. The unit digit of 812m² + 3 is equal to
Answer : 1

Question. In ΔABC, the vertex A = (1, 2), y = x is the perpendicular bisector of the side AB and x – 2y + 1 = 0 is the equation of the internal angle bisector of L . If the equation of the side BC is ax + by – 5 = 0, then the value of a – b is ..... .
Answer : 4

Question. A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. If the point O divides the segment PQ in the ratio m/n , then m + n is ________.
Answer : 7

Question. The vertex of an equilateral triangle is (2, –1), and the equation of its base is x + 2y = 1. If the length of its sides is 2 /√K , then value of K is ____.
Answer : 15

Question. If (sin θ, cos θ), θ ∈ [0, 2π] and (1, 4) lie on the same side or on the line √3x – y + 1 = 0, then the maximum value of sin θ will be ______.
Answer : 0

Question. The straight lines (3secq + 5cosec q)x + (7sec θ - 3cosecθ)y +11(secθ - cosecθ) = 0 always pass through a fixed point P for all possible values of q. If the maximum value of the difference of distances of P and B (3, 4) from a point on the line x - y + 3 = 0 is k then k²/10 is equal to .
Answer : 4

Question. If tana, tanb, tanl are the roots of the equation t³ – 12t² + 15t – 1 = 0; then the centroid of triangle having vertices (tana, cota); (tanβ, cotbb); (tanλ, cotl) is given by G(h, k); then evaluate (h + k)/(k – h).
Answer : 9

Question. Consider a ΔABC whose sides AB, BC, and CA are represented by the straight lines 2x + y = 0, x + py = q, and x – y = 3, respectively. The point P(2, 3) is the orthocenter. The value of (p + q)/10 is ..... .
Answer : 5