Mathematics · Textbook solutions

Straight Lines

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 94 questions

9.2 Slope of a Line

15 q

Solved Examples

Worked · 3
  1. 9.1 Eg.1
    Find the slope of the lines: (a) Passing through the points (3,−2)(3, -2) and (−1,4)(-1, 4), (b) Passing through the points (3,−2)(3, -2) and (7,−2)(7, -2), (c) Passing through the points (3,−2)(3, -2) and (3,4)(3, 4), (d) Making inclination of 60∘60^\circ with the positive direction of xx-axis.
  2. 9.1 Eg.2
    If the angle between two lines is π4\frac{\pi}{4} and slope of one of the lines is 12\frac{1}{2}, find the slope of the other line.
  3. 9.1 Eg.3
    Line through the points (−2,6)(-2, 6) and (4,8)(4, 8) is perpendicular to the line through the points (8,12)(8, 12) and (x,24)(x, 24). Find the value of xx.

Exercise 9.1

Practice · 12
  1. Ex 9.1 Q1
    Draw a quadrilateral in the Cartesian plane, whose vertices are (−4,5)(-4, 5), (0,7)(0, 7), (5,−5)(5, -5) and (−4,−2)(-4, -2). Also, find its area.
  2. Ex 9.1 Q2
    The base of an equilateral triangle with side 2a2a lies along the yy-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.
  3. Find the distance between P(x1,y1)P(x_1, y_1) and Q(x2,y2)Q(x_2, y_2) when :
    Ex 9.1 Q3(i)
    PQ is parallel to the yy-axis.
  4. Ex 9.1 Q3(ii)
    PQ is parallel to the xx-axis.
  5. Ex 9.1 Q4
    Find a point on the xx-axis, which is equidistant from the points (7,6)(7, 6) and (3,4)(3, 4).
  6. Ex 9.1 Q5
    Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P(0,−4)P(0, -4) and B(8,0)B(8, 0).
  7. Ex 9.1 Q6
    Without using the Pythagoras theorem, show that the points (4,4)(4, 4), (3,5)(3, 5) and (−1,−1)(-1, -1) are the vertices of a right angled triangle.
  8. Ex 9.1 Q7
    Find the slope of the line, which makes an angle of 30∘30^\circ with the positive direction of yy-axis measured anticlockwise.
  9. Ex 9.1 Q8
    Without using distance formula, show that points (−2,−1)(-2, -1), (4,0)(4, 0), (3,3)(3, 3) and (−3,2)(-3, 2) are the vertices of a parallelogram.
  10. Ex 9.1 Q9
    Find the angle between the xx-axis and the line joining the points (3,−1)(3, -1) and (4,−2)(4, -2).
  11. Ex 9.1 Q10
    The slope of a line is double of the slope of another line. If tangent of the angle between them is 13\frac{1}{3}, find the slopes of the lines.
  12. Ex 9.1 Q11
    A line passes through (x1,y1)(x_1, y_1) and (h,k)(h, k). If slope of the line is mm, show that k−y1=m(h−x1)k - y_1 = m(h - x_1).

9.3 Various Forms of the Equation of a Line

24 q

Solved Examples

Worked · 5
  1. 9.2 Eg.4
    Find the equations of the lines parallel to axes and passing through (−2,3)(-2, 3).
  2. 9.2 Eg.5
    Find the equation of the line through (−2,3)(-2, 3) with slope −4-4.
  3. 9.2 Eg.6
    Write the equation of the line through the points (1,−1)(1, -1) and (3,5)(3, 5).
  4. 9.2 Eg.7
    Write the equation of the lines for which tan⁡θ=12\tan\theta = \frac{1}{2}, where θ\theta is the inclination of the line and (i) yy-intercept is −32-\frac{3}{2} (ii) xx-intercept is 44.
  5. 9.2 Eg.8
    Find the equation of the line, which makes intercepts −3-3 and 22 on the xx- and yy-axes respectively.

Exercise 9.2

Practice · 19
  1. Ex 9.2 Q1
    Write the equations for the xx-and yy-axes.
  2. Ex 9.2 Q2
    Find the equation of the line which satisfies the given condition: Passing through the point (−4,3)(-4, 3) with slope 12\frac{1}{2}.
  3. Ex 9.2 Q3
    Find the equation of the line which satisfies the given condition: Passing through (0,0)(0, 0) with slope mm.
  4. Ex 9.2 Q4
    Find the equation of the line which satisfies the given condition: Passing through (2, 23)\left(2,\ 2\sqrt{3}\right) and inclined with the xx-axis at an angle of 75∘75^\circ.
  5. Ex 9.2 Q5
    Find the equation of the line which satisfies the given condition: Intersecting the xx-axis at a distance of 3 units to the left of origin with slope −2-2.
  6. Ex 9.2 Q6
    Find the equation of the line which satisfies the given condition: Intersecting the yy-axis at a distance of 2 units above the origin and making an angle of 30∘30^\circ with positive direction of the xx-axis.
  7. Ex 9.2 Q7
    Find the equation of the line which satisfies the given condition: Passing through the points (−1,1)(-1, 1) and (2,−4)(2, -4).
  8. Ex 9.2 Q8
    The vertices of Δ\Delta PQR are P (2,1)(2, 1), Q (−2,3)(-2, 3) and R (4,5)(4, 5). Find equation of the median through the vertex R.
  9. Ex 9.2 Q9
    Find the equation of the line passing through (−3,5)(-3, 5) and perpendicular to the line through the points (2,5)(2, 5) and (−3,6)(-3, 6).
  10. Ex 9.2 Q10
    A line perpendicular to the line segment joining the points (1,0)(1, 0) and (2,3)(2, 3) divides it in the ratio 1:n1 : n. Find the equation of the line.
  11. Ex 9.2 Q11
    Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2,3)(2, 3).
  12. Ex 9.2 Q12
    Find equation of the line passing through the point (2,2)(2, 2) and cutting off intercepts on the axes whose sum is 9.
  13. Ex 9.2 Q13
    Find equation of the line through the point (0,2)(0, 2) making an angle 2π3\frac{2\pi}{3} with the positive xx-axis. Also, find the equation of line parallel to it and crossing the yy-axis at a distance of 2 units below the origin.
  14. Ex 9.2 Q14
    The perpendicular from the origin to a line meets it at the point (−2,9)(-2, 9), find the equation of the line.
  15. Ex 9.2 Q15
    The length L (in centimetre) of a copper rod is a linear function of its Celsius temperature C. In an experiment, if L=124.942L = 124.942 when C=20C = 20 and L=125.134L = 125.134 when C=110C = 110, express L in terms of C.
  16. Ex 9.2 Q16
    The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs 17/litre?
  17. Ex 9.2 Q17
    P (a,b)(a, b) is the mid-point of a line segment between axes. Show that equation of the line is xa+yb=2\frac{x}{a} + \frac{y}{b} = 2.
  18. Ex 9.2 Q18
    Point R (h,k)(h, k) divides a line segment between the axes in the ratio 1:21 : 2. Find equation of the line.
  19. Ex 9.2 Q19
    By using the concept of equation of a line, prove that the three points (3,0)(3, 0), (−2,−2)(-2, -2) and (8,2)(8, 2) are collinear.

9.4 Distance of a Point From a Line

24 q

Solved Examples

Worked · 2
  1. 9.3 Eg.9
    Find the distance of the point (3,−5)(3, -5) from the line 3x−4y−26=03x - 4y - 26 = 0.
  2. 9.3 Eg.10
    Find the distance between the parallel lines 3x−4y+7=03x - 4y + 7 = 0 and 3x−4y+5=03x - 4y + 5 = 0.

Exercise 9.3

Practice · 22
  1. Reduce the following equations into slope - intercept form and find their slopes and the yy - intercepts.
    Ex 9.3 Q1(i)
    x+7y=0x + 7y = 0
  2. Ex 9.3 Q1(ii)
    6x+3y−5=06x + 3y - 5 = 0
  3. Ex 9.3 Q1(iii)
    y=0y = 0
  4. Reduce the following equations into intercept form and find their intercepts on the axes.
    Ex 9.3 Q2(i)
    3x+2y−12=03x + 2y - 12 = 0
  5. Ex 9.3 Q2(ii)
    4x−3y=64x - 3y = 6
  6. Ex 9.3 Q2(iii)
    3y+2=03y + 2 = 0
  7. Ex 9.3 Q3
    Find the distance of the point (−1,1)(-1, 1) from the line 12(x+6)=5(y−2)12(x + 6) = 5(y - 2).
  8. Ex 9.3 Q4
    Find the points on the xx-axis, whose distances from the line x3+y4=1\frac{x}{3} + \frac{y}{4} = 1 are 4 units.
  9. Find the distance between parallel lines
    Ex 9.3 Q5(i)
    15x+8y−34=015x + 8y - 34 = 0 and 15x+8y+31=015x + 8y + 31 = 0
  10. Ex 9.3 Q5(ii)
    l(x+y)+p=0l(x + y) + p = 0 and l(x+y)−r=0l(x + y) - r = 0
  11. Ex 9.3 Q6
    Find equation of the line parallel to the line 3x−4y+2=03x - 4y + 2 = 0 and passing through the point (−2,3)(-2, 3).
  12. Ex 9.3 Q7
    Find equation of the line perpendicular to the line x−7y+5=0x - 7y + 5 = 0 and having xx intercept 3.
  13. Ex 9.3 Q8
    Find angles between the lines 3x+y=1\sqrt{3}x + y = 1 and x+3y=1x + \sqrt{3}y = 1.
  14. Ex 9.3 Q9
    The line through the points (h,3)(h, 3) and (4,1)(4, 1) intersects the line 7x−9y−19=07x - 9y - 19 = 0. at right angle. Find the value of hh.
  15. Ex 9.3 Q10
    Prove that the line through the point (x1,y1)(x_1, y_1) and parallel to the line Ax+By+C=0Ax + By + C = 0 is A(x−x1)+B(y−y1)=0A(x - x_1) + B(y - y_1) = 0.
  16. Ex 9.3 Q11
    Two lines passing through the point (2,3)(2, 3) intersects each other at an angle of 60∘60^\circ. If slope of one line is 2, find equation of the other line.
  17. Ex 9.3 Q12
    Find the equation of the right bisector of the line segment joining the points (3,4)(3, 4) and (−1,2)(-1, 2).
  18. Ex 9.3 Q13
    Find the coordinates of the foot of perpendicular from the point (−1,3)(-1, 3) to the line 3x−4y−16=03x - 4y - 16 = 0.
  19. Ex 9.3 Q14
    The perpendicular from the origin to the line y=mx+cy = mx + c meets it at the point (−1,2)(-1, 2). Find the values of mm and cc.
  20. Ex 9.3 Q15
    If pp and qq are the lengths of perpendiculars from the origin to the lines xcos⁡θ−ysin⁡θ=kcos⁡2θx\cos\theta - y\sin\theta = k\cos 2\theta and xsec⁡θ+y cosec θ=kx\sec\theta + y\,\mathrm{cosec}\,\theta = k, respectively, prove that p2+4q2=k2p^2 + 4q^2 = k^2.
  21. Ex 9.3 Q16
    In the triangle ABC with vertices A (2,3)(2, 3), B (4,−1)(4, -1) and C (1,2)(1, 2), find the equation and length of altitude from the vertex A.
  22. Ex 9.3 Q17
    If pp is the length of perpendicular from the origin to the line whose intercepts on the axes are aa and bb, then show that 1p2=1a2+1b2\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}.

Miscellaneous Exercise on Chapter 9

31 q

Miscellaneous Examples

Worked · 6
  1. Misc Eg.11
    If the lines 2x+y−3=02x + y - 3 = 0, 5x+ky−3=05x + ky - 3 = 0 and 3x−y−2=03x - y - 2 = 0 are concurrent, find the value of kk.
  2. Misc Eg.12
    Find the distance of the line 4x−y=04x - y = 0 from the point P (4,1)(4, 1) measured along the line making an angle of 135∘135^\circ with the positive xx-axis.
  3. Misc Eg.13
    Assuming that straight lines work as the plane mirror for a point, find the image of the point (1,2)(1, 2) in the line x−3y+4=0x - 3y + 4 = 0.
  4. Misc Eg.14
    Show that the area of the triangle formed by the lines y=m1x+c1y = m_1x + c_1, y=m2x+c2y = m_2x + c_2 and x=0x = 0 is (c1−c2)22∣m1−m2∣\frac{(c_1 - c_2)^2}{2|m_1 - m_2|}.
  5. Misc Eg.15
    A line is such that its segment between the lines 5x−y+4=05x - y + 4 = 0 and 3x+4y−4=03x + 4y - 4 = 0 is bisected at the point (1,5)(1, 5). Obtain its equation.
  6. Misc Eg.16
    Show that the path of a moving point such that its distances from two lines 3x−2y=53x - 2y = 5 and 3x+2y=53x + 2y = 5 are equal is a straight line.

Miscellaneous Exercise

Practice · 25
  1. Find the values of kk for which the line (k−3)x−(4−k2)y+k2−7k+6=0(k-3)x - (4 - k^2)y + k^2 - 7k + 6 = 0 is
    Misc Q1(a)
    Parallel to the xx-axis,
  2. Misc Q1(b)
    Parallel to the yy-axis,
  3. Misc Q1(c)
    Passing through the origin.
  4. Misc Q2
    Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and −6-6, respectively.
  5. Misc Q3
    What are the points on the yy-axis whose distance from the line x3+y4=1\frac{x}{3} + \frac{y}{4} = 1 is 4 units.
  6. Misc Q4
    Find perpendicular distance from the origin to the line joining the points (cos⁡θ, sin⁡θ)(\cos\theta,\ \sin\theta) and (cos⁡ϕ, sin⁡ϕ)(\cos\phi,\ \sin\phi).
  7. Misc Q5
    Find the equation of the line parallel to yy-axis and drawn through the point of intersection of the lines x−7y+5=0x - 7y + 5 = 0 and 3x+y=03x + y = 0.
  8. Misc Q6
    Find the equation of a line drawn perpendicular to the line x4+y6=1\frac{x}{4} + \frac{y}{6} = 1 through the point, where it meets the yy-axis.
  9. Misc Q7
    Find the area of the triangle formed by the lines y−x=0y - x = 0, x+y=0x + y = 0 and x−k=0x - k = 0.
  10. Misc Q8
    Find the value of pp so that the three lines 3x+y−2=03x + y - 2 = 0, px+2y−3=0px + 2y - 3 = 0 and 2x−y−3=02x - y - 3 = 0 may intersect at one point.
  11. Misc Q9
    If three lines whose equations are y=m1x+c1y = m_1x + c_1, y=m2x+c2y = m_2x + c_2 and y=m3x+c3y = m_3x + c_3 are concurrent, then show that m1(c2−c3)+m2(c3−c1)+m3(c1−c2)=0m_1(c_2 - c_3) + m_2(c_3 - c_1) + m_3(c_1 - c_2) = 0.
  12. Misc Q10
    Find the equation of the lines through the point (3,2)(3, 2) which make an angle of 45∘45^\circ with the line x−2y=3x - 2y = 3.
  13. Misc Q11
    Find the equation of the line passing through the point of intersection of the lines 4x+7y−3=04x + 7y - 3 = 0 and 2x−3y+1=02x - 3y + 1 = 0 that has equal intercepts on the axes.
  14. Misc Q12
    Show that the equation of the line passing through the origin and making an angle θ\theta with the line y=mx+cy = mx + c is yx=m±tan⁡θ1∓mtan⁡θ\frac{y}{x} = \frac{m \pm \tan\theta}{1 \mp m\tan\theta}.
  15. Misc Q13
    In what ratio, the line joining (−1,1)(-1, 1) and (5,7)(5, 7) is divided by the line x+y=4x + y = 4?
  16. Misc Q14
    Find the distance of the line 4x+7y+5=04x + 7y + 5 = 0 from the point (1,2)(1, 2) along the line 2x−y=02x - y = 0.
  17. Misc Q15
    Find the direction in which a straight line must be drawn through the point (−1,2)(-1, 2) so that its point of intersection with the line x+y=4x + y = 4 may be at a distance of 3 units from this point.
  18. Misc Q16
    The hypotenuse of a right angled triangle has its ends at the points (1,3)(1, 3) and (−4,1)(-4, 1). Find an equation of the legs (perpendicular sides) of the triangle which are parallel to the axes.
  19. Misc Q17
    Find the image of the point (3,8)(3, 8) with respect to the line x+3y=7x + 3y = 7 assuming the line to be a plane mirror.
  20. Misc Q18
    If the lines y=3x+1y = 3x + 1 and 2y=x+32y = x + 3 are equally inclined to the line y=mx+4y = mx + 4, find the value of mm.
  21. Misc Q19
    If sum of the perpendicular distances of a variable point P (x,y)(x, y) from the lines x+y−5=0x + y - 5 = 0 and 3x−2y+7=03x - 2y + 7 = 0 is always 10. Show that P must move on a line.
  22. Misc Q20
    Find equation of the line which is equidistant from parallel lines 9x+6y−7=09x + 6y - 7 = 0 and 3x+2y+6=03x + 2y + 6 = 0.
  23. Misc Q21
    A ray of light passing through the point (1,2)(1, 2) reflects on the xx-axis at point A and the reflected ray passes through the point (5,3)(5, 3). Find the coordinates of A.
  24. Misc Q22
    Prove that the product of the lengths of the perpendiculars drawn from the points (a2−b2, 0)\left(\sqrt{a^2 - b^2},\ 0\right) and (−a2−b2, 0)\left(-\sqrt{a^2 - b^2},\ 0\right) to the line xacos⁡θ+ybsin⁡θ=1\frac{x}{a}\cos\theta + \frac{y}{b}\sin\theta = 1 is b2b^2.
  25. Misc Q23
    A person standing at the junction (crossing) of two straight paths represented by the equations 2x−3y+4=02x - 3y + 4 = 0 and 3x+4y−5=03x + 4y - 5 = 0 wants to reach the path whose equation is 6x−7y+8=06x - 7y + 8 = 0 in the least time. Find equation of the path that he should follow.