(18) Find the ratio in which the join of the points A(12, –5) andB(3, 4) is cut by the line r (3i – 11j) = 0. Show that the linejoining the origin to the point dividing AB externally in thesame ratio is perpendicular to the line r - (3i – 11j) = 0.(19) Find the equations of the sides of the square which has the point(-1, 1) as one vertex and the line r (3i + j) – 3 = 0 for onediagonal.(20) The points A(3, –1), B(-2, 4), C(-1,7) are three vertices ofthe quadrilateral ABCD. Given that the triangle ADC is thereflection in the line AC of triangle ABC, find the positionvector of D.(21) Find the reflection of the point (2, 1) in the line with equationr: (5i – j) + 3 = 0. Hence find the reflection in L of the liner:(i - j) –1 = 0.(22) Find the mirror images of the point B(3, 4) and the line3i + 4j + t(13i + j) in the line r (3i +7j) +2 = 0. (1) Find the perpendicular distances of(a) the point (-6, 2) from the line r (5i+ 12j) = 7;(b) the point (3, 4) from the line r - (3i + 4j) = 10;(e) the point (2, -3) from the line r (i – 5j) – 1 = 0;(d) the origin from the line r - (7ị – 3j) – 29 = 0.(2) Find the position vector of of the foot of the perpendicular fromthe point with position vector đị – 2j to the line r-(2i – 3) = 1.(3) Find the length of the altitude of the triangle A(2, 1), B(–1,3), C(-4, –3)from B to AC.(4) If Q is the foot of the perpendicular from (8,7) to the linejoining A(3, 6) to B(9, 2), find the position vector of Q, and theratio in which Q divides AB.(5) P and Q are the feet of the perpendiculars from A(0, 5) andB(-6,0) to the line r:(-i+2;) = 4. Find(a) the equations of AP and BQ.the distance PQ,(c) the position vector of the point of intersection of AQ withthe r-axis.(6) Find the distances between the following pairs of parallel lines:(a) r - (i + j) + 7 = 0, r · (i +¿) – 11 = 0;(b) r- (2i – 3) + 6 = 0, r · (di – 6) +5 = 0;(c) r • (4i – 3) – 9 = 0, r · (di – 3) – 24 = 0;(d) r = i+j+s(4i –i). r = 4ị + 5j +t(8į – 25)(7) Find the equations of the bisectors of the angles between thefollowing pairs of lines and, in each case, discriminate betweenthem:(a) r- (2i + j) + 4 = 0, r · (2i – 4j) – 7 = 0;(b) r- (3i + 4j) + 5 = 0, r - (5i + 12j) + 13 = 0;(d) r- (3i + 4) – 12 = 0, r - (3i – j) – 5 = 0;(e) r- (24i +7j) = 20, r- (di – 3) = 2.(8) Find the equation of the bisectors of the angles between the linejoining the points with position vectors -3i – 9j. 2i + 3j andthe line meeting the r- and y- axes at the points with positionvectors 3i and 4j.12(9) Find the equation of the line, which is such that the axis of 1bisects the angle between it and the line r · (2i + 5j) = 18.(10) If y = mx bisects an angle between the lines y = m,1, y = m3a,show that(m – m,)(1+ mm,) + (m – m,)(1+ mm,) = 0,and deduce that the two bisectors are given by the roots of theequation(m, + ma)m² + 2(1 – m,mą]m – (m, + m2) = 0.Show that the two values of m given by the above equationdetermine perpendicular lines. Show also that if m, = 1/m,then the bisectors are y =1, y = -1.(11) The position vectors of the vertices of a triangle area = -i+j. b = Ti – 2j. c = 5i + 3j.Find the length of one side of the triangle and the length of thealtitude to that side and hence compute the area of the triangle.

Note: Since you have asked multiple questions, we will solve the first question for you which is in…

(18) Find the ratio in which the join of the points A(12, –5) and B(3,4) is cut by the line r (3i – 11j) = 0. Show that the line joining the origin to the point dividing AB externally in the same ratio is perpendicular to the line r- (3i – 11j) = 0.

(18) Find the ratio in which the join of the points A(12, –5) and B(3,4) is cut by the line r (3i – 11j) = 0. Show that the line joining the origin to the point dividing AB externally in the same ratio is perpendicular to the line r- (3i – 11j) = 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.

Question

(18) Find the ratio in which the join of the points A(12, –5) and
B(3, 4) is cut by the line r (3i – 11j) = 0. Show that the line
joining the origin to the point dividing AB externally in the
same ratio is perpendicular to the line r - (3i – 11j) = 0.
(19) Find the equations of the sides of the square which has the point
(-1, 1) as one vertex and the line r (3i + j) – 3 = 0 for one
diagonal.
(20) The points A(3, –1), B(-2, 4), C(-1,7) are three vertices of
the quadrilateral ABCD. Given that the triangle ADC is the
reflection in the line AC of triangle ABC, find the position
vector of D.
(21) Find the reflection of the point (2, 1) in the line with equation
r: (5i – j) + 3 = 0. Hence find the reflection in L of the line
r:(i - j) –1 = 0.
(22) Find the mirror images of the point B(3, 4) and the line
3i + 4j + t(13i + j) in the line r (3i +7j) +2 = 0.

(1) Find the perpendicular distances of
(a) the point (-6, 2) from the line r (5i+ 12j) = 7;
(b) the point (3, 4) from the line r - (3i + 4j) = 10;
(e) the point (2, -3) from the line r (i – 5j) – 1 = 0;
(d) the origin from the line r - (7ị – 3j) – 29 = 0.
(2) Find the position vector of of the foot of the perpendicular from
the point with position vector đị – 2j to the line r-(2i – 3) = 1.
(3) Find the length of the altitude of the triangle A(2, 1), B(–1,3), C(-4, –3)
from B to AC.
(4) If Q is the foot of the perpendicular from (8,7) to the line
joining A(3, 6) to B(9, 2), find the position vector of Q, and the
ratio in which Q divides AB.
(5) P and Q are the feet of the perpendiculars from A(0, 5) and
B(-6,0) to the line r:(-i+2;) = 4. Find
(a) the equations of AP and BQ.
the distance PQ,
(c) the position vector of the point of intersection of AQ with
the r-axis.
(6) Find the distances between the following pairs of parallel lines:
(a) r - (i + j) + 7 = 0, r · (i +¿) – 11 = 0;
(b) r- (2i – 3) + 6 = 0, r · (di – 6) +5 = 0;
(c) r • (4i – 3) – 9 = 0, r · (di – 3) – 24 = 0;
(d) r = i+j+s(4i –i). r = 4ị + 5j +t(8į – 25)
(7) Find the equations of the bisectors of the angles between the
following pairs of lines and, in each case, discriminate between
them:
(a) r- (2i + j) + 4 = 0, r · (2i – 4j) – 7 = 0;
(b) r- (3i + 4j) + 5 = 0, r - (5i + 12j) + 13 = 0;
(d) r- (3i + 4) – 12 = 0, r - (3i – j) – 5 = 0;
(e) r- (24i +7j) = 20, r- (di – 3) = 2.
(8) Find the equation of the bisectors of the angles between the line
joining the points with position vectors -3i – 9j. 2i + 3j and
the line meeting the r- and y- axes at the points with position
vectors 3i and 4j.
1
2
(9) Find the equation of the line, which is such that the axis of 1
bisects the angle between it and the line r · (2i + 5j) = 18.
(10) If y = mx bisects an angle between the lines y = m,1, y = m3a,
show that
(m – m,)(1+ mm,) + (m – m,)(1+ mm,) = 0,
and deduce that the two bisectors are given by the roots of the
equation
(m, + ma)m² + 2(1 – m,mą]m – (m, + m2) = 0.
Show that the two values of m given by the above equation
determine perpendicular lines. Show also that if m, = 1/m,
then the bisectors are y =1, y = -1.
(11) The position vectors of the vertices of a triangle are
a = -i+j. b = Ti – 2j. c = 5i + 3j.
Find the length of one side of the triangle and the length of the
altitude to that side and hence compute the area of the triangle.

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