Find the distances between the following pair of points 117. (i) (b + c, c + a) and (c + a, a + b) (ii) (a cos, a sin) and (a cos, a sin) (iii) (am 1, am 1 ) and (am, am ). 118. (i) Find the value of x 1 if the distance between the points (x 1, ) and (3, 4) be 8. (ii) Prove that the points (a, 4a), (a, 6a) and (a + 3a,5a) are the vertices of an equilateral triangle whose side is a. (iii) Prove that the points (, 1), (1, 0), (4, 3) and (1, ) are at the vertices of a parallelogram. 1 9 119. Prove that the point, is the centre of the circle circumscribing the triangle whose angular points are (1, ), (, 3) and (, ). 10. Find the coordinates of the points which (i) Divides the line joining the points (1, 3) and (, 7) in the ratio 3 : 4. (ii) Divides, internally and externally, the line joining ( 1, ) to (4, 5) in the ratio : 3. 11. (i) The line joining the points (1, ) and ( 3, 4) is trisected; find the coordinates of the points of trisection. (ii) Find the points which divide the line segment joining (8, 1) and (1, 8) into four equal parts. (iii) If (, 1), (1, 0) and (4, 3) are three successive vertices of parallelogram, find the fourth vertex. 1. Show that the following points, taken in order, form the figure mentioned against the points. (i) (, 5), (3, 4), (7, 10)... Right angled isosceles triangle. (ii) (1, 3), (3, 1), ( 5, 5)... Right angled triangle. (iii) ( 3, 1), ( 6, 7), (3, 9), (6, 1)... Parallelogram. (iv) (8, 4), (5, 7), ( 1, 1), (, )... Rectangle. (v) (3, ), (7, 6), ( 1, ), ( 5, 6)... Rhombus. 13. Find the area of the triangle having vertices (i) (0, 4), (3, 6) and ( 8, ). (ii) (5, ), ( 9, 3) and ( 3, 5). (iii) (a, c + a), (a, c) and ( a, c a). (iv) (a cos 1, b sin 1 ), (a cos, b sin ) and (a cos 3, b sin 3 ). (v) (am 1, am 1 ), (am, am ) and (am 3, am 3 ). (vi) {am 1 m, a(m 1 + m )}, {am m 3, a(m + m 3 )} and (am 3 m 1, a(m 3 + m 1 )} a a a (vii) am 1,, am, and am 3,. m1 m m3 14. Prove that the following sets of three points are in a straight line: (i) (1, 4), (3, ), and ( 3, 16). 1 (ii),3,( 5,6 ) and ( 8, 8). (iii) (a, b + c), (b, c + a), and (c, a + b) 15. Find the area of the quadrilaterals the coordinates of whose angular points, taken in order, are: (i) (1, 1), (3, 4), (5, ) and (4, 7). (ii) ( 1, 6), ( 3, 9), (5, 8) and (3, 9). 16. (i) Find the value of k, if the area of the triangle formed by (k, 0), (3, 4) and (5, ) is 10 sq. units. (ii) Show that the following points are collinear (0, ), ( 1, 1), (, 4).
(iii) Find the value of k if the points (k, 1), (, 1) and (4, 5) are collinear. 17. (i) Find the centroid of the triangle whose vertices are (, 7), (3, 1) and ( 5, 6). (ii) Find the In centre of the triangle, whose vertices are A(3, ), B(7, ) and C(7, 5). 18. (i) Find the coordinates of (1, ) with respect to new axes, when the origin is shifted to (, 3) by translation of axes. (ii) Find the transformed equation of x + 4xy + 5y = 0, when the origin is shifted to (3, 4) by the translation of axes. (iii) The coordinates of a point are changed as ( 4, 3), when the origin is shifted to (1, 5) by the translation of axes. Find the coordinates of the point in the original system. (iv) When the origin is shifted to (4, 5) by the translation of axes, find the coordinates of the following points with reference to new axes. (a) (0, 3) (b) (, 4) (c) (4, 5) 19. (i) Find the point to which the origin is to be shifted by translation of axes so as to remove the first degree terms from the equation ax + hxy + by + gx + fy + c = 0, where h ab. (ii) Find the point to which the origin is to be shifted by the translation of axes so as to remove the first degree terms from the equation ax + by + gx + fy + c = 0, where a 0, b 0. (iii) Find the point to which the origin is to be shifted so as to remove the first degree terms from the equation 4x + 9y 8x + 36y + 4 = 0. 130. (i) Find the point to which the origin is to be shifted so that the point (3, 0) may change to (, 3). (ii) Find the transformed equations of the following when the origin is shifted to ( 1, ) by translation of axes. (a) x + y + x 4y + 1 = 0 (b) x + y 4x + 4y = 0 (iii) The point to which the origin is shifted and the transformed equation are given below. Find the original equation. (a) (3, 4); x + y = 4 (b) ( 1, ); x + y + 16 = 0 (iv) If the transformed equation of a curve is x + 3xy y + 17x 17y 11 = 0, when the origin is shifted to the point (, 3), then find the original equation of the curve. 131. (i) Find the equation of the straight line making an angle of 10 with the positive direction of the X axis and passing through the point (0, ). (ii) Find the equation of the straight line which makes an angle 135 with the positive X axis and passes through the point (, 3). (iii) Write the equations of the straight lines parallel to Y axis and (i) at a distance of units from the Y axis to the right of it, (ii) at a distance of 5 units from the Y axis to the left of it. (iv) Find the value of x, if the slope of the line passing through (, 5) and (x, 3) is. 13. Find the equations of the straight lines which makes the following angles with the positive X axis in the positive direction and which pass through the points given below. (i) 4 and (0, 0) (ii) 4 and (1, ) (iii) 135 and (3, ) (iv) 150 and (, 1) 133. (i) Find the equations of the straight lines passing through the origin and making equal angles with the coordinate axes. (ii) Find the equation of the straight line passing through ( 4, 5) and cutting off equal nonzero intercepts on the coordinate axes. (iii) Find the equation of the straight line passing through (, 4) and making nonzero intercepts whose sum is zero. (iv) Find the equation of the straight line passing through the point (4, 3) and perpendicular to the line passing through the points (1, 1) and (, 3). 134. Find the equation of the straight line passing through
a a (i) (a cos 1, a sin 1 ) and (a cos, a sin ) (ii) at 1, andat, t1 t (iii) (at 1, at 1 ) and (at, at ) 135. (i) Find the equation to the straight line which passes through the point ( 4, 3) and is such that the portion of it between the axes is divided by the point in the ratio 5 : 3. (ii) Find the equation to the straight lines which passes through the point (1, ) and cut off equal intercept from the axes. 136. In what follows, p denotes the distance of the straight line from the origin. The normal ray drawn from the origin to the straight line makes an angle with the positive direction of the X axis. Find the equations of the straight lines with the following values of p and. (i) p = 5, = 60 (ii) p = 6, = 150 (iii) p = 3/, = /3 (iv) p = 1, = 7/4 (v) p = 4, = 90 (vi) p =.5, = 0 137. Find the equations of the straight lines in the symmetric form, in the following cases having the given slope and passing through the given point. (i) Slope = 3, point (, 3) (ii) Slope = 1, point (, 0) (iii) Slope = 1, point (1, 1) 3 138. (i) Reduce to the perpendicular form the equation x + y3 + 7 = 0. (ii) A straight line parallel to the line y = 3x passes through Q(, 3) and cuts the line x + 4y 7 = 0 at P. Find the length of PQ. (iii) Find the points on the line 4x 3y 10 = 0 which are at a distance of 5 units from the point (1, ). (iv) Transform the equation 5x y 7 = 0 into (i) slope intercept form (ii) intercept form and (iii) normal form 139. (a) (i) Find the value of k, if the straight lines 6x 10y + 3 = 0 and kx 5y + 8 = 0 are parallel. (ii) Find the value of p, if the straight lines 3x + 7y 1 = 0 and 7x py + 3 = 0 are mutually perpendicular. (iii) Find the value of k, so that the straight lines y 3kx + 4 = 0 and (k 1)x (8k 1)y 6 = 0 are perpendicular. (b) (iv) The line x y 1meets the X axis at P. Find the equation of the line perpendicular to this line at P. Find the angles between the pairs of straight lines: (i) x y3 = 5 and 3x + y = 7 (ii) x 4y = 3 and 6x y = 11 (iii) y = 3x + 7 and 3y x = 8 (iv) y = ( 3)x + 5 and y = ( + 3)x 7 (v) (m mn)y = (mn + n )x + n 3 and (mn + m )y = (mn n )x + m 3 140. Find the equation to the straight line passing through the point ( 6, 10) and perpendicular to the straight line 7x + 8y = 5. 141. Find the equation to the straight line (i) Passing through the point (, 3) and perpendicular to the straight line 4x 3y = 10. (ii) Passing through the point (, 3) and perpendicular to the straight line joining the points (5, 7) and ( 6, 3). (iii) Passing through the point ( 4, 3) and perpendicular to the straight line joining (1, 3) and (, 7)
14. (i) Prove that the equation to the straight line which passes through the point (a cos 3, a sin 3 ) and is perpendicular to the straight line x sec + y cosec = a is x cos y sin = a cos. (ii) Find the equations to the straight lines passing through (x, y) and respectively perpendicular to the straight lines (a) xx + yy = a xx' yy', (b) 1, 143. Find the angle between the two straight lines 3x = 4y + 7 and 5y = 1x + 6 and also the equations to the two straight lines which pass through the point (4, 5) and make equal angles with the two given lines. 144. (i) Find the equations to the straight lines which pass through the origin and are inclined at 75 to the straight line x + y + 3 (y x) = a (ii) Find the equations to the straight lines which pass through the point (h, k) and are inclined at an angle tan 1 m to the straight line y = mx + c. 145. (i) Find the values of k, if the angle between the straight lines kx + y + 9 = 0 and y 3x = 4 is 45 (ii) Find the equations of the straight lines passing through the point ( 10, 4) and making an angle with the line x y = 10 such that tan =. 146. Find the equations of the straight lines passing through the point (1, ) and making an angle of 60 with the line 3x + y = 0. 147. Find the length of the perpendicular drawn from (i) The point (4, 5) upon the straight line 3x + 4y = 10. (ii) The origin upon the straight line x y 1. 3 4 (iii) The point ( 3, 4) upon the straight line 1(x + 6) = 5(y ). (iv) The point (b, a) upon the straight line x y 1. (v) Find the length of the perpendicular from the origin upon the straight line joining the two points whose coordinates are (a cos, a sin ) and (a cos, a sin ) 148. Show that the product of the perpendiculars drawn from the two points ( a b, 0) upon the straight line cos sin 1is b. 149. (i) Find the distance between the two parallel straight lines y = mx + c and y = mx + d (ii) What are the points on the axis of x whose perpendicular distance from the straight line 1 is a? 150. (i) Find the distance between parallel straight lines 3x + 4y 3 = 0 and 6x + 8y 1 = 0. (ii) Find the distance between the following parallel lines: (a) 3x 4y = 1, 3x 4y = 7 (b) 5x 3y 4 = 0, 10x 6y 9 = 0 151. Find the orthocentre of the triangle whose vertices are ( 5, 7), (13, ) and ( 5, 6). 15. (i) Find the length of the perpendicular drawn from the point given against the following straight lines. (a) 5x y + 4 = 0... (, 3) (b) 3x 4y + 10 = 0... (3, 4) (c) x 3y 4 = 0... (0, 0) (ii) Find the perpendicular distance from the point ( 6, 5) to the straight lines 5x 1y = 3. 153. (i) Find the foot of the perpendicular drawn from (4, 1) upon the straight line 3x 4y + 1 = 0 (ii) Find the foot of the perpendicular drawn from ( 1, 3) on the line 5x y = 18.
(iii) Find the foot of the perpendicular drawn from (3, 0) upon the straight line 5x + 1y 14 = 0 154. (i) Find the image of the point (1, ) w.r.t the straight line 3x + 4y 1 = 0 (ii) Find the image of (, 3) w.r.t the straight line 4x 5y + 8 = 0 (iii) x 3y 5 = 0 is the perpendicular bisector of the line segment joining the points A, B. If A = ( 1, 3), find the coordinates of B. (iv) Prove that the feet of the perpendiculars from the origin on the lines x + y = 4, x + 5y = 6 and 15x 7y = 44 all lie on a straight line. 155. Show that the distance of the point (6, ) from the line 4x + 3y = 1 is half the distance of the point (3, 4) from the line 4x 3y = 1. 156. Each side of a square is of length 4 units. The centre of the square is (3, 7) and one of its diagonals is parallel to y = x. Find the coordinates of its vertices. 157. Find the equations to the straight lines bisecting the angles between the following pairs of straight lines, placing first the bisector of the angle in which the origin lies. (i) 1x + 5y 4 = 0 and 3x + 4y + 7 = 0. (ii) 4x + 3y 7 = 0 and 4x + 7y 31 = 0. (iii) x + y = 4 and y + 3x = 5. 158. Find the equations to the bisectors of the internal angles of the triangles the equations of whose sides are respectively (i) 3x + 4y = 6, 1x 5y = 3, and 4x 3y + 1 = 0. (ii) 3x + 5y = 15, x + y = 4, and x + y = 6. 159. (i) Find the direction in which a straight line must be drawn through the point (1, ) so that its point of intersection with the line x + y = 4 may be at a distance of 1 6 from this point. 3 (ii) Find the equation of the straight line passing through the point of intersection of the lines x + y + 1 = 0 and x y + 5 = 0 and containing the point (5, ). 160. (i) Find the value of k, if the lines x 3y + k = 0, 3x 4y 13 = 0 and 8x 11y 33 = 0 are concurrent. (iii) If the straight lines ax + by + c = 0, bx + cy + a = 0 and cx + ay + b = 0 are concurrent, then prove that a 3 + b 3 + c 3 = 3abc. (iv) Show that the lines x + y 3 = 0, 3x + y = 0 and x 3y 3 = 0 are concurrent and find the point of concurrency. (v) Show that the straight line (a b)x + (b c)y = c a, (b c)x + (c a)y = a b and (c a)x + (a b)y = b c are concurrent. 161. (i) Find the value of p, if the following lines are concurrent. (a) 3x + 4y = 5, x + 3y = 4, px + 4y = 6 (b) 4x 3y 7 = 0, x + py + = 0, 6x + 5y 1 = 0 (ii) If 3a + b + c = 0, then show that the equation ax + by + c = 0 represents a family of concurrent straight lines and find the point of concurrency. 16. (i) Show that the straight lines x + y = 0, 3x + y 4 = 0 and x + 3y 4 = 0 form an isosceles triangle. (ii) Find the area of the triangle formed by the straight lines x y 5 = 0, x 5y + 11 = 0 and x + y 1 = 0. 163. (i) Find the locus of the third vertex of a right angled triangle, the ends of whose hypotenuse are (4, 0) and (0, 4). (ii) Find the equation of the locus of a point which is equidistant from the points A( 3, ) and B(0, 4).
164. (i) Find the equation of the locus of point P such that the distance of P from the origin is twice the distance of P from A(1, ). (ii) A(, 3) and B( 3, 4) be two given points. Find the equation of the locus of P so that the area of the triangle PAB is 8.5 sq. units. 165. (i) Find the equation of locus of a point P, if the distance of P from A (3, 0) is twice the distance of P from B( 3, 0). (ii) Find the equation of locus of a point which is equidistant from the coordinate axes. 166. (i) Find the equation of the locus of a point P, the square of whose distance from the origin is 4 times its y coordinate. (ii) Find the equation of locus of a point equidistant from A (, 0) and the Y axis. 167. (i) Find the equation of locus of P, if the ratio of the distances from P to (5, 4) and (7, 6) is : 3.
(ii)find the equation of locus of P, if the line segment joining (, 3) and ( 1, 5) subtends a right angle at P. 168. The ends of the hypotenuse of a right angled triangle are (0, 6) and (6, 0). Find the equation of locus of its third vertex. 169. (i) Find the equation of locus of a point the difference of whose distance from ( 5, 0) and (5, 0) is 8 units. (ii) Find the equation of locus of P, if A = (4, 0), B = ( 4, 0) and PA + PB = 4. 170. (i) Find the equation of locus of a point, the sum of whose distances from (0, ) and (0, ) is 6 units. (ii) Find the equation of locus of P, if A = (, 3), B = (, 3) and PA + PB = 8. 171. (i) A (5, 3) and B (3, ) are two fixed points. Find the equation of locus of P, so that the area of triangle PAB is 9 units. (ii) Find the equation of locus of the point P such that PA + PB = c, where A = (a, 0), B = ( a, 0) and 0 < a < c. 17. If the distance from P to the points (, 3) and (, 3) are in the ratio : 3, then find the equation of locus of P. 173. A (1, ), B (, 3) and C (, 3) are three points. A point P moves such that PA + PB = PC. Find the equation to the locus of P. 117. (i) (a b) (b c) (ii) asin (iii) a(m m 1) (m1 m ) 4 10 33 16 118. (i) x 1 = 3 15 10. (i), (ii) 1,, ( 11, 4) 7 7 5 1 5 11. (i),0, (ii) (9, 11) (10, 10) (11, 9) (iii) (1, ) 3 3 13. (i) 1 (ii) 9 (iii) a (iv) ab sin sin sin (v) a (m m 3 )(m 3 m 1 )(m 1 m ) (vi) ½ a (m 1 m )(m m 3 )(m 3 m 1 ) (vii) ½ a (m 1 m )(m m 3 )(m 3 m 1 ) + m 1 m m 3 15. (i) 0½ (ii) 96 16. (i) k = 1 or k = 3 3 (iii) k = 1 17. (i) (0, 4) (ii) (6, 3) 18. (i) (3, 1) (ii) x + 4xy + 5y + 8x + 5y + 146 = 0 (iii) ( 3, 8) (iv) (a) ( 4, 8) (b) ( 6, 9) (c) (0, 0) hf bg gh af 19. (ii), ab h ab h (iii) (, ) (1, ) g f (ii), a b (iv) 1 7, 3 3 1 1 130. (i) (1, 3) (ii) x + y 4 = 0 (iii) x + y + x 8y + 5 = 0 (iv) x y + 3xy + 4x 11y + 10 = 0 131. (i) y = 3x (ii) y = x + 1 (iii) x =, x = 5 (iv) x = 1
13. (i) y = x (ii) x y + 1 = 0 (iii) x + y = 1 (iv) x + 3 y + ( + 3 ) = 0 133. (i) y = x (ii) x + y = 1 (iii) x y + 6 = 0 (iv) x + y + = 0 134. (i) 1 1 1 xcos y sin acos (ii) t 1 t y + x = a(t 1 + t ) (iii) y(t 1 + t ) x = at 1 t 135. (i) 0y 9x = 96 (ii) x + y + 1 = 0 136. (i) x + 3 y 10 = 0 (ii) 3 x y + 1 = 0 (iii) x 3 y + 3 = 0 (iv) x y = 0 (v) y = 4 (vi) x = 5/ 1 137. (i) 3 (x ) = (y 3) (ii) y 0 = (x + ) (iii) y + x = 3 138. (i) x cos40 + y sin40 = 7 1 (ii) PQ = (iii) x = 4, y = 1 3 (iv) (a) y = 5 x 7, 139. (a) (i) k = 3, (ii) p = 3, (iii) k = 1 or k = 1/6, (iv) ax + by = a (b) (i), (ii) _ 1 3 tan 10, (iii) _ 1 4 tan 4m n, (iv) 60, (v) tan 3 m n _ 1 4 4 140. 8x 7y + 118 = 0 141. (i) 3x + 4y = 18 (ii) 11x + 4y = 10 (iii) x + 4y + 16 = 0 14. (ii) (a) yx xy = 0 (b) a yx b xy = (a b )(xy) 143. 1 33 tan, 7x + 9y = 17, 7y 9x + 1 = 0 56 144. (i) y + 3 x = 0, x = 0 (ii) y = k, y k = m (x h) 1 m 145. (i) k =, 1/ (ii) x + 10 = 0, 3x + 4y + 14 = 0 146. y =, y = 3 (x 1) 147. (i) 4 5, (ii) 5, (iii) 1 5 13, (iv) c d a 149. (i) b,0 1 m b 1 150. (i) ½, (ii) (a) 1, (b) 17 15. (i) (a) 0, (b) 3 5, (c) 4 10, (ii) a ab b a b 151. ( 3, ) (ii) 93 13, (v) acos
8 1 05 1 7 6 153. (i), (ii) (4, ), (iii), 154. (i), 5 5 169 169 5 5, (ii) 6 10, 15 3 (iii) 8 6, 5 5 156. (1, 5) (1, 9) (5, 8) (5, 5) 159. (i) = 15 or 75 (ii) 3x + 7y 1 = 0 160. (i) k = 7, (ii) (4, 5) 161. (i) (a), (b) 4, (ii) (3, ) 16. (ii) 9 sq. units. 163. (i) x + y 4x 4y = 0 (ii) 6x + 4y = 3 164. (i) 3x + 3y 8x 16y + 0 = 0 (ii) x + 5y = 0, x + 5y = 34 165. (i) x + y + 10x + 9 = 0 (ii) x y = 0 166. (i) x + y 4y = 0 (ii) y 4x + 4 = 0 167. (i) 5(x + y ) 34x + 10y + 9 = 0 (ii) x + y x 8y + 13 = 0 168. x + y 6x 6y = 0 169. (i) 1 (ii) 16 9 170. (i) 1 (ii) 16x + 7y 64x 48 = 0 5 9 171. (i) (5x y 37)(5x y 1) = 0 (ii) x + y = c a 1 16 16 3 17. 5x + 5y 0x 78y + 65 = 0 173. 7x 7y + 4 = 0