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The angle between the lines √3x – y – 2 = 0 and x – √3y + 1 = 0 is

1) 90o 2) 60o 3) 45o 4) 30o Solution: Given lines are √3x - y - 2 = 0 …(i) x - √3y + 1 = 0 …(ii) Slope of line (i) m1 = √3 Slope of line... View Article

If (a, a²) falls inside the angle made by the lines y = x/2, x > 0 and y = 3x, x > 0, then a belongs to

1) (3, ∞) 2) (1/2, 3) 3) (-3, - (1/2)) 4) (0, 1/2) Solution: The given lines are y = x/2 and y = 3x. The point (a, a2) lies inside the... View Article

The angle between the lines represented by the equation 2x² + 3xy – 5y² = 0, is

1) π/3 2) π/2 3) tan-1 |12/5| 4) tan-1 (7/3) Solution: Given 2x2 + 3xy - 5y2 = 0 Comparing this equation with the general equation ax2 +... View Article

The angle between the pair of lines (x² + y²) sin² α = (x cos θ – y sin θ)² is

1) θ 2) 2θ 3) α 4) 2α Solution: Given (x2 + y2) sin2 α = (x cos θ - y sin θ)2 => x2 sin2 α + y2 sin2 α - x2 cos2 θ - y2 sin2 θ + 2xy sin... View Article

The angle between the pair of straight lines formed by joining the points of intersection of x² + y² = 4 and y = 3x + c to the origin is a right angle. Then, c² is equal to

1) 20 2) 13 3) 1/5 4) 5 Solution: Given x2 + y2 = 4 …(i) Also y = 3x + c => (y - 3x)/c = 1 …(ii) (homogenization) Rearranging (i) x2 + y2... View Article

A line passes through the point of intersection of the lines 100x + 50y – 1 = 0 and 75x + 25y + 3 = 0 and makes an equal intercept on the axes. Its equation is

1) 25x + 25y - 1 = 0 2) 5x - 5y + 3 = 0 3) 25x + 25y - 4 = 0 4) 25x - 25y + 6 = 0 Solution: Given equation of lines are 100x + 50y - 1 =... View Article

The lines (a + 2b)x + (a – 3b)y = a – b for different values of a and b pass through the fixed point, whose coordinates are

1) (2/5, 3/5) 2) (3/5, 3/5) 3) (1/5, 1/5) 4) (3/5, 2/5) Solution: Given equation of line is (a + 2b)x + (a - 3b)y = a - b => ax + 2bx +... View Article

A line passes through point (2, 2) and perpendicular to the line 3x + y = 3. Then, y – intercept is

1) 2 2) 5/3 3) 4/3 4) 3/4 Solution: Given line is 3x + y = 3 => y = -3x + 3 Slope = -3 So the slope of the perpendicular line = 1/3... View Article

The pair of lines joining origin to the points of intersection of the two curves ax² + 2hxy + by² + 2gx = 0 and a’x²+ 2h’xy + b’y² + 2g’x = 0 will be at right angles, if

1) (a' + b' ) g' = (a + b) g 2) (a + b ) g' = (a' + b') g 3) h2 - ab = h'2 - a'b' 4) a + b + h2 = a' + b' + h'2 Solution: Given equation of... View Article

If θ is the angle between the lines ax² + 2bxy + by² = 0, then angle between x² + 2xy sec θ + y² = 0 is

1) θ 2) 2θ 3) θ/2 4) 3θ Solution: Given equation of lines is x2 + 2xy sec θ + y2 = 0 Comparing with the general equation, we get a = 1, h =... View Article

If the line px² – qxy – y² = 0 makes the angles α and β with X – axis, then the value of tan (α + β) is

(1) -q/(1+p) (2) q/(1+p) (3) -q/(1-p) (4) none of these Solution: Given α and β are the angles with X - axis. So slope, m1 = tan α m2 = tan... View Article

All chords of the curve 3x² – y² – 2x + 4y = 0 which subtend a right angle at the origin, pass through the fixed point

1) (1, 2) 2) (1, - 2) 3) (-1, 2) 4) (-1, -2) Solution: Given curve is 3x2 - y2 - 2x + 4y = 0 …(i) Let the equation of the chord is y = mx +... View Article

The length of the perpendicular from the point (a cos α, a sin α) upon the straight line y = x tan α + c (where c > 0) is

1) c 2) c sin2 α 3) c cos α 4) c sec2 α Solution: Given y = x tan α + c => x tan α - y + c = 0 Given point is (a cos α, a sin α) The... View Article

If the equation of base of an equilateral triangle is 2x – y = 1 and the vertex (-1, 2), then the length of the side of the triangle is

1) √(20/3) 2) 2/√15 3) 5 4) √5 Solution: Let A be the point (-1,2). Since the triangle is equilateral, angle B = 600. BD represents... View Article

The distance between the pair of parallel lines x² + 2xy + y² – 8ax – 8ay – 9a² = 0 is

1) 2√5 a 2) √10 a 3) 10a 4) 5√2 a Solution: Given x2 + 2xy + y2 - 8ax - 8ay - 9a2 = 0 Comparing above equation with general second degree... View Article

If the foot of the perpendicular from the origin to a straight line is at the point (3, -4 ). Then, the equation of the line is

1) 3x - 4y = 25 2) 3x - 4y + 25 = 0 3) 4x + 3y - 25 = 0 4) 4x + 3y + 25 = 0 Solution: Let P be the foot of the perpendicular from origin to... View Article

A line through the point A (2, 0) which makes an angle of 30⁰ with the positive direction of X-axis is rotated about A in clockwise direction through an angle of 15⁰. Then, the equation of the straight line in the new position is

1) (2 - √3) x + y + 4 + 2 √3 = 0 2) (2 - √3)x - y - 4 + 2 √3 = 0 3) (2 - √3)x + y + 4 + 2 √3 = 0 4) (2 - √3)x + y + 4 + 2 √3 = 0 Solution:... View Article

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