Question 1

Consider the following statements:

Statement 1: The greatest possible number of intersection points formed by 8 circles of the same radius is 56.

Statement 2: The greatest possible number of points of intersection obtained when 4 circles of different radii and 4 distinct straight lines a

Accepted Answer

Statement 1 is correct, Statement 2 is correct, but Statement 2 does not correctly explain Statement 1. 8 equal circles: each pair meets in at most 2 points, giving 2*C(8,2) = 56, so Statement 1 is correct. For 4 circles + 4 lines: circle-circle 2*C(4,2)=12, line-line C(4,2)=6, line-circle 4*4*2=32, total = 50, so Statement 2 is also correct, but it does not explain Statement 1.

Question 2

Find the length of the segment from the point (1, 2) to the line x + y + 5 = 0, if the measurement is taken along a line parallel to 3x − y = 7.

Accepted Answer

√40. Move from (1,2) along direction (1,3): (1+t, 2+3t). Hitting x+y+5=0 gives 3+4t+5=0, t = -2. Distance = |t|*sqrt(1+9) = 2*sqrt(10) = sqrt(40).

Question 3

Let p1 and p2 denote the perpendicular distances from the origin to the lines x cos θ + y sin θ = 2a cos 4θ and x sec θ + y cosec θ = 4a cos 2θ, respectively. If mp1² + np2² = 4a², then which of the following is true?

Accepted Answer

m = 1, n = 4. For the first line p1=2a*cos4t (already normalized), so p1^2=4a^2 cos^2(4t). For the second, p2=4a*cos2t/sqrt(sec^2 t+cosec^2 t)=2a*cos2t*sin2t=a*sin4t, so p2^2=a^2 sin^2(4t). Then m*p1^2+n*p2^2=4a^2 means 4m cos^2(4t)+n sin^2(4t)=4 for all t, giving m=1, n=4.

Question 4

For which value of p² does the equation y² + xy + px² − x − 2y + p = 0 represent a pair of straight lines?

Accepted Answer

1/4. For a pair of lines, the determinant condition with a=p, b=1, h=1/2, g=-1/2, f=-1, c=p gives p^2 - (5/4)p + 1/4 = 0 -> p=1/4 or p=1. The valid value listed is 1/4.

Question 5

An equilateral triangle has one vertex at (2, 3). If the side opposite to this vertex lies on the line x + y = 2, then the equations of the other two sides are

Accepted Answer

y - 3 = (2 ± √3)(x - 2). The base x+y=2 has slope -1. Lines through the vertex (2,3) making 60 deg with it satisfy sqrt(3) = |(m+1)/(1-m)|, giving m = 2 +/- sqrt(3). Thus y - 3 = (2 +/- sqrt(3))(x - 2).

Question 6

Which points on the line x + y = 4 are exactly one unit away from the line 4x + 3y = 10?

Accepted Answer

(3, 1), (−7, 11). The points (3, 1) and (−7, 11) are both located on the line x + y = 4 and are exactly one unit away from the line 4x + 3y = 10, as verified by calculating the perpendicular distance from these points to the line, which equals one unit.

Question 7

The line y = x - 2 is turned about the point where it meets the x-axis until it becomes perpendicular to the line ax + by + c = 0. What is the resulting equation of the line?

Accepted Answer

ay - bx + 2b = 0. y=x-2 meets the x-axis at (2,0). Perpendicular to ax+by+c=0 (slope -a/b) needs slope b/a. The line through (2,0) with slope b/a is bx - ay - 2b = 0, i.e. ay - bx + 2b = 0.

Question 8

How many distinct straight lines can be drawn through the point (2, 3) such that, together with the coordinate axes, they enclose a triangle of area 12 square units?

Accepted Answer

3. For line x/a + y/b = 1 through (2,3) with area (1/2)|ab| = 12, so |ab| = 24 and 2b+3a = ab. With ab = 24: (a-4)^2 = 0 gives one line (a=4). With ab = -24: a^2+8a-16 = 0 gives two real solutions. Total 3 lines.

Question 9

For the pair of straight lines given by x² + 2hxy + 2y² = 0, if the ratio of their slopes is 1: 2, what is the value of h?

Accepted Answer

±3/2. The slopes of the lines represented by the equation can be derived from the coefficients of the quadratic terms. Given the ratio of the slopes is 1:2, we can set up an equation involving h, leading to the conclusion that h must equal ±3/2 to satisfy this ratio.

Question 10

What is the perpendicular distance from the point (-1, 3) to the line 2x + y = 3, measured along the direction of a line whose slope is 1?

Accepted Answer

2√2/3. From (-1,3) along (1,1): (-1+t, 3+t). Substituting into 2x+y=3: 2(-1+t)+(3+t)=3 -> 3t = 2, t = 2/3. Distance = |t|*sqrt(2) = 2sqrt(2)/3.

Question 11

A straight line cuts the coordinate axes, and the segment between the axes is divided by the point (−5, 4) in the ratio 1:2. Which equation represents this line?

Accepted Answer

−8x + 5y = 60. The point (−5, 4) divides the segment between the x-intercept and y-intercept in the ratio 1:2, allowing us to derive the equation of the line using the section formula. The correct equation, −8x + 5y = 60, satisfies the conditions of the intercepts and the given ratio.

Question 12

Find the image of the point (4, -13) when reflected across the line 5x + y + 6 = 0.

Accepted Answer

(-1, -14). Foot-of-perpendicular reflection: d=(5*4+(-13)+6)/(25+1)=13/26=1/2. Image x=4-2*5*(1/2)=-1, y=-13-2*1*(1/2)=-14. Image is (-1,-14).

Question 13

Find the combined equation of the two straight lines passing through the point (1, 0) and parallel to the pair of lines given by 2x² - xy - y² = 0.

Accepted Answer

2x² - xy - y² - 4x + y + 2 = 0. The correct option represents the combined equation of two lines that are parallel to the original pair of lines defined by the quadratic equation and also pass through the specified point (1, 0). By substituting the point into the equation, it confirms that the lines intersect at that point, thus validating the solution.

Question 14

What is the equation of the line representing the hour hand at 4 o’clock?

Accepted Answer

x + √3y = 0. At 4 o'clock the hour hand is 120 deg clockwise from 12 (the +y axis), i.e. at -30 deg from the +x axis, with slope tan(-30) = -1/sqrt(3). The line through the origin is y = -x/sqrt(3), i.e. x + sqrt(3)y = 0.

Question 15

A line L reflects point P(2, 3) to Q(4, 5). Under reflection in the same line L, what is the image of the point R(0, 0)?

Accepted Answer

(7, 7). L is the perpendicular bisector of PQ: midpoint (3,4), slope -1, so x+y=7. Reflecting (0,0) in x+y=7 gives (7,7).

Question 16

Find the coordinates of the point that lies at a distance of +3 from the point (1, -3) on the line 2x + 3y + 7 = 0.

Accepted Answer

(1 + 9/√13, -3 - 6/√13). The line 2x+3y+7=0 has direction (3,-2)/sqrt(13). From (1,-3), the point at +3 along it is (1 + 9/sqrt(13), -3 - 6/sqrt(13)). The option with y-coordinate 1 - 9/sqrt(13) is impossible.

Question 17

A square has one diagonal lying on the line x = 2y. If one of its vertices is at (3, 0), then the equations of the two sides passing through this vertex are

Accepted Answer

y - 3x + 9 = 0, 3y + x - 3 = 0. The diagonal x=2y has slope 1/2; the sides through a vertex make 45 degrees with it, giving slopes (1/2+1)/(1-1/2)=3 and (1/2-1)/(1+1/2)=-1/3. Through (3,0): y-3x+9=0 and 3y+x-3=0 (these are perpendicular, as required).

Question 18

For the one-parameter family of lines a(2x + y + 4) + b(x - 2y - 3) = 0, how many lines in the family are at a distance √10 from the point P(2, -3)?

Accepted Answer

1. The equation represents a family of lines parameterized by a and b. To find the number of lines at a specific distance from point P(2, -3), we can derive the distance formula from the line equation and set it equal to the given distance, resulting in a quadratic equation that has one solution, indicating that there is exactly one line at the specified distance.

Question 19

A line parallel to the x-axis passes through the point of intersection of the lines ax + 2by + 3b = 0 and bx - 2ay - 3a = 0, where (a, b) ≠ (0, 0). This line is

Accepted Answer

below the x-axis at a distance of 3/2 units. The correct option is right because the intersection of the given lines can be calculated, revealing that the y-coordinate of the intersection point is -3/2, which indicates that the line parallel to the x-axis at this point is indeed below the x-axis at a distance of 3/2 units.

Question 20

The quadratic equation 8x² + 8xy + 2y² + 26x + 13y + 15 = 0 is the equation of two straight lines. What is the separation between these two lines?

Accepted Answer

7/(2√5). 8x^2+8xy+2y^2 = 2(2x+y)^2 and 26x+13y = 13(2x+y). Let u=2x+y: 2u^2+13u+15=0 gives u=-3/2 or u=-5, i.e. parallel lines 2x+y=-3/2 and 2x+y=-5. Distance = |(-3/2)-(-5)|/sqrt(2^2+1^2) = (7/2)/sqrt5 = 7/(2sqrt5).

Question 21

A line L passes through the point (3, −2) and makes an angle of 60° with the line √3x + y = 1. If L also cuts the x-axis, then which of the following is its equation?

Accepted Answer

y − √3x + 2 + 3√3 = 0. Angle 60 deg with line of slope -sqrt(3) gives slopes 0 or sqrt(3). Slope 0 (y=-2) never meets the x-axis, so take slope sqrt(3) through (3,-2): y - sqrt(3)x + 2 + 3 sqrt(3) = 0.

Question 22

Which of the following is the equation of a line that passes through the point (0, a) and has perpendicular distance a from the point (2a, 2a)?

Accepted Answer

Both (a) and (b). Both candidate lines pass through (a,0) and lie at distance a from (2a,2a): for 3x-4y-3a=0 the distance is |6a-8a-3a|/5 = a, and for x=a the distance is |2a-a| = a. Hence both equations satisfy the condition.

Question 23

The coordinates (1,3) and (5,1) represent a pair of opposite corners of a rectangle. If the other two corners lie on the straight line y = 2x + c, then one of those corners is

Accepted Answer

(4,4). Centre = midpoint (3,2); the line through the other corners passes through it, so c=-4 (y=2x-4). Intersecting with the circle (x-3)^2+(y-2)^2=5 gives (2,0) and (4,4); (4,4) is listed.

Question 24

Find the shortest distance between the two parallel lines 6x + 8y + 15 = 0 and 3x + 4y + 9 = 0.

Accepted Answer

3/10 unit. Write 6x+8y+15=0 as 3x+4y+7.5=0. Distance between parallel lines = |7.5-9|/sqrt(3^2+4^2) = 1.5/5 = 3/10.

Question 25

A triangle has two vertices fixed at P(a,0) and Q(0,b), while the third vertex R(x,y) moves on the line y = x. If the area of the triangle is A, then the value of dA/dx is

Accepted Answer

−(a + b)/2. With P(a,0), Q(0,b), R(x,x), area = (1/2)(ab - ax - bx). Differentiating with respect to x gives dA/dx = -(a+b)/2.

Question 26

Consider the three lines L1: a1x + b1y + c1 = 0, L2: a2x + b2y + c2 = 0, and L3: a3x + b3y + c3 = 0. Statement-1: If these three lines are concurrent, then the determinant |a1 b1 c1; a2 b2 c2; a3 b3 c3| equals 0. Statement-2: If |a1 b1 c1; a2 b2 c2; a3 b3 c3| equals 0, then L1, L2, and L3

Accepted Answer

Statement-1 is true, Statement-2 is false.. If three lines are concurrent, the coefficient determinant is 0 (Statement-1 true). The converse fails: det = 0 can also occur when two or all three lines are parallel/coincident, so they need not be concurrent (Statement-2 false). Hence S1 true, S2 false.

Question 27

ABCD is a rectangle-shaped plot. A lamp post of height 12 m is fixed vertically at corner A. If the angle subtended by the top of the post at B is 60° and at C is 45°, then the area of the plot is

Accepted Answer

48√2 sq.m. From B the elevation 60 gives AB=12/tan60=4sqrt3; from C the elevation 45 gives the diagonal AC=12. Then the other side AD=sqrt(AC^2-AB^2)=sqrt(144-48)=4sqrt6, so area = AB*AD = 4sqrt3 * 4sqrt6 = 48sqrt2 sq.m.

Question 28

Find the set of all real values of m such that the two points P and Q lie on the line y = mx + 8 and satisfy ∠APB = ∠AQB = π/2, where A = (-4, 0) and B = (4, 0).

Accepted Answer

(-∞, -√3) ∪ (√3, ∞) {-2, 2}. The correct option identifies the slopes of the line formed by points P and Q that create right angles with the line segment AB. The conditions for the angles being π/2 lead to restrictions on the slope m, specifically excluding values between -√3 and √3, as well as the slopes -2 and 2, which correspond to the perpendicularity condition.

Question 29

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is 1/4. Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man?

Accepted Answer

B only. Line x/a + y/b = 1 with mean of reciprocal intercepts (1/a+1/b)/2 = 1/4 gives 1/a+1/b = 1/2. A point (k,k) lies on it when k(1/a+1/b)=1 -> k(1/2)=1 -> k=2. So only (2,2), stone B, is on the path.

Question 30

ABCD is a trapezium such that AB and CD are parallel and AB ⟂ CD. If ∠ADB = θ, BC = p and CD = q, then AB is equal to:

Accepted Answer

(p² + q²) sinθ / (p cosθ + q sinθ). Using the right triangle relations in the trapezium, AB = (p^2 + q^2) sin(theta) / (p cos(theta) + q sin(theta)). The denominator appears to the first power, not squared.

Question 31

For the triangle whose vertices are at (4, 0), (-1, -1), and (3, 5), the triangle is

Accepted Answer

isosceles and right-angled. The triangle has two sides of equal length, making it isosceles, and the angles formed by these sides include a right angle, confirming that it is also right-angled.

Question 32

Find the locus of the midpoint of the segment cut off by the coordinate axes from the line x cos α + y sin α = p, where p is a constant.

Accepted Answer

1/x² + 1/y² = 4/p². The correct option represents the locus of the midpoint of the segment cut off by the coordinate axes from the given line. By finding the intercepts and calculating the midpoint, we derive the relationship that leads to the equation 1/x² + 1/y² = 4/p², which describes a rectangular hyperbola.

Question 33

If the pair of straight lines represented by ax² + 2hxy + by² + 2gx + 2fy + c = 0 meet on the y-axis, then which relation must hold?

Accepted Answer

2fgh = bg² + ch². The condition for the pair of straight lines to meet on the y-axis implies that the x-coefficient must vanish, leading to the relationship 2fgh = bg² + ch², which ensures that the lines intersect at a point where x equals zero.

Question 34

For what values of a does the equation 3ax² + 5xy + (a² − 2)y² = 0 represent a pair of straight lines that are mutually perpendicular?

Accepted Answer

for two distinct values of a. The equation represents a pair of straight lines that are mutually perpendicular when the discriminant condition is satisfied, which leads to two distinct values of 'a' that fulfill this requirement.

Question 35

A square of side a is placed in the upper half-plane with one corner at the origin. The side through the origin forms an angle α, where 0 < α < π/4, with the positive x-axis. Find the equation of the diagonal that does not pass through the origin.

Accepted Answer

y(cos α + sin α) + x(cos α − sin α) = a. The correct option represents the equation of the diagonal by combining the contributions of both the x and y coordinates, adjusted for the angle α. The terms reflect the orientation of the square in the upper half-plane, ensuring that the diagonal correctly accounts for the angle formed with the axes.

Question 36

If x₁, x₂, x₃ and y₁, y₂, y₃ each form geometric progressions with the same common ratio, then the points (x₁, y₁), (x₂, y₂), and (x₃, y₃)

Accepted Answer

are collinear. The points (x₁, y₁), (x₂, y₂), and (x₃, y₃) are collinear because both sets of coordinates form geometric progressions with the same common ratio, which implies that the ratios of the differences between corresponding x and y coordinates are constant, satisfying the condition for collinearity.

Question 37

The equation of the straight line passing through the point (4, 3) and making intercepts on the co-ordinate axes whose sum is −1 is

Accepted Answer

x/2 − y/3 = 1 and x/−2 + y/1 = 1. The correct option represents lines that pass through the point (4, 3) and have intercepts on the axes that sum to -1, satisfying both the intercept condition and the point condition.

Question 38

If the sum of the slopes of the lines given by x² − 2cxy − 7y² = 0 is four times their product, c has the value

Accepted Answer

2. The relationship between the sum and product of the slopes of the lines represented by the quadratic equation can be derived from the coefficients of the equation. In this case, setting the sum of the slopes equal to four times their product leads to a specific value for c, which is found to be 2.

Question 39

If one of the lines given by 6x² − xy + 4cy² = 0 is 3x + 4y = 0, then c equals

Accepted Answer

−3. Write 6x^2 - xy + 4cy^2 = (3x+4y)(2x+qy). Matching gives 4q = 4c so q = c, and the xy-term 3q+8 = -1 -> 3c = -9 -> c = -3.

Question 40

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) ≠ (0, 0), is

Accepted Answer

below the x-axis at a distance of 3/2 from it. Solving ax+2by+3b=0 and bx-2ay-3a=0 gives the intersection (0, -3/2). The line through it parallel to the x-axis is y = -3/2, which lies below the x-axis at a distance 3/2 from it.

Question 41

If vertex of a triangle is (1, 1) and the mid points of two sides through this vertex are (−1, 2) and (3, 2) then the centroid of the triangle is

Accepted Answer

(1, 7/3). The centroid of a triangle is calculated as the average of the coordinates of its vertices. Given the vertex at (1, 1) and the midpoints at (−1, 2) and (3, 2), the centroid is found by averaging these points, resulting in (1, 7/3).

Question 42

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

Accepted Answer

4x + 3y = 24. If A(3,4) bisects the intercept, the axis intercepts are (6,0) and (0,8). The line x/6 + y/8 = 1 gives 4x + 3y = 24.

Question 43

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

Accepted Answer

(1/2, 3). The point (a, a²) must lie between the lines y = x/2 and y = 3x for positive x values. For a to satisfy this condition, its corresponding y-value a² must be greater than x/2 and less than 3x, which leads to the conclusion that a must be in the interval (1/2, 3).

Question 44

Let A(1, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which 'k' can take is given by

Accepted Answer

{−1, 3}. The area of triangle ABC can be calculated using the formula for the area of a triangle, which is 1/2 * base * height. Given that AC is the hypotenuse, the height can be determined by the vertical distance from point B to line AC, and the base is the horizontal distance between points A and C. Solving the area equation leads to the values of k being -1 and 3.

Question 45

Given the points P = (-1, 0), Q = (0, 0), and R = (3, 3√(3)), find the equation of the angle bisector of ∠PQR.

Accepted Answer

√(3)x + y = 0. The correct option represents the line that bisects the angle formed by the lines connecting points P and Q, and Q and R. By calculating the slopes of these lines and finding the angle bisector, we determine that the equation √(3)x + y = 0 correctly describes this bisector.

Question 46

For the pair of straight lines represented by my² + (1-m²)xy - mx² = 0, if one line is an angle bisector of the two coordinate axes given by xy = 0, then the value of m is

Accepted Answer

1. The given equation represents two straight lines, and for one of them to be an angle bisector of the coordinate axes, it must have a slope of 1 or -1. Substituting m = 1 into the equation results in one of the lines having a slope of 1, confirming that it bisects the angle between the axes.

Question 47

The perpendicular bisector of the segment joining P(1, 4) and Q(k, 3) cuts the y-axis at -4. Which of the following can be a value of k?

Accepted Answer

-4. The perpendicular bisector of a segment is a line that is equidistant from both endpoints and intersects the y-axis at the midpoint's y-coordinate. For the segment joining P(1, 4) and Q(k, 3), the midpoint's y-coordinate must equal -4, which occurs when k is -4, making it the only valid option.

Question 48

For which value(s) of p do the lines p(p²+1)x - y + q = 0 and (p² + 1)x + (p² + 1)y + 2q = 0 both stand perpendicular to the same line?

Accepted Answer

exactly one value of p. The lines are perpendicular to the same line if their slopes are negative reciprocals of each other. This condition leads to a unique relationship between the coefficients of the lines, resulting in exactly one value of p that satisfies this requirement.

Question 49

Let A(2, -3) and B(-2, 1) be two vertices of a triangle. If the third vertex is allowed to move along the line 2x + 3y = 9, then the path traced by the centroid of the triangle is:

Accepted Answer

2x + 3y = 1. The centroid of a triangle is the average of the coordinates of its vertices. As the third vertex moves along the line 2x + 3y = 9, the centroid's coordinates can be expressed as a linear combination of the fixed vertices A and B, leading to the equation 2x + 3y = 1.

Question 50

The straight line 2x + y = k goes through the point that divides the segment joining (1,1) and (2,4) internally in the ratio 3:2. What is the value of k?

Accepted Answer

6. To find the value of k, we first determine the coordinates of the point that divides the segment joining (1,1) and (2,4) in the ratio 3:2 using the section formula. This gives us the point (1.6, 2.6). Substituting these coordinates into the equation of the line 2x + y = k, we find that k equals 6.

JEE Main Maths: Straight Lines questions with solutions

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