Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3

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1 Drill Exercise Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ). 2. Prove that the points (2a, 4a) (2a, 6a) and (2a + 3 a, 5a) are the vertices of an equilateral triangle. 3. Which point on y-axis is equidistant from (2, 3) and ( 4, 1)? Drill Exercise Find the ratio in which the line segment joining (2, 3) and (5, 6) is divided by (i) x-axis (ii) y-axis. 2. If three vertices of a parallelogram are (a + b, a b), (2a + b, 2a b), (a b, a + b), then find the fourth vertex. 3. Find the coordinates of points on the line joining the points P(3, 4) and Q( 2, 5) that is twice as far from P as from Q. Drill Exercise Find the orthocentre of the triangle formed by the lines xy = 0 and x + y = In a triangle ABC with vertices A(1, 2), B(2, 3) and C(3, 1) and A = cos 1, 5 B = C = cos then find the circumcentre of the triangle ABC. 3. If G be the centroid and I be the incentre of the triangle with vertices A( 36, 7), B(20, 7) and 25 C(0, 8) and GI = then find the value of. Drill Exercise The vertices of ABC are ( 2, 1), (5, 4) and (2, 3) respectively. Find the area of the triangle and the length of the altitude through A. 2. Prove that the points (a, 0), (0, b) and (1, 1) are collinear if a 1 + b 1 = The four vertices of a quadrilateral are (1, 2), ( 5, 6), (7, 4) and (k, 2) taken in order. If the area of the quadrilateral is zero, find the value of k.

2 Drill Exercise Find the locus of a point equidistant from the point (2, 4) and the y-axis. 2. Find the locus of a point, so that the join of ( 5, 1) and (3, 2) subtends a right angle at the moving point. 3. If O is the origin and Q is a variable point on y 2 = x. Find the locus of the mid-point of OQ. Drill Exercise (a) Determine x so that the line passing through (3, 4) and (x, 5) makes 135º angle with the positive direction of x-axis. (b) If the line passing through the points (2, 5), ( 5, 5) then prove that line will be parallel to x-axis and if the line passing through the points (6, 3), (6, 3) then prove that line will be perpendicular to the x-axis. 2. If is the angle of inclination of the line joining the points (7, 2) and (3, 1), then find the value of sin and cos. 3. Using the method of slope, show that the following points are collinear (i) A(4, 8), B(5, 12), C(9, 28) (ii) A(16, 18), B (3, 6), C( 10, 6) Drill Exercise Are the points (3, 4) and (2, 6) on the same or opposite sides of the line 3x 4y = 8? 2. Which one of the points (1, 1), ( 1, 2) and (2, 3) lies on the side of the line 4x + 3y 5 = 0 on which the origin lies. 3. If the points (4, 7) and (cos, sin ), where 0 < <, lie on the same side of the line x + y 1 = 0, then prove that lies in the first quadrant. Drill Exercise A straight line is drawn through the point P(2, 3) and is inclined at an angle of 30º with the x-axis. Find the coordinates of two points on it at a distance 4 from P on either side of P. 2. Find the distance of the point (2, 3) from the line 2x 3y + 9 = 0 measured along a line x y + 1 = Find the equation of the line which passes through P(1, 7) and meets the axes at A and B respectively so that 4AP 3BP = 0, where O is the origin. Drill Exercise Show that if any line through the variable point A(k + 1, 2k), meets the lines 7x + y 16 = 0, 5x y 8 = 0, x 5y + 8 = 0 at B, C, D respectively AC, AB and AD are in harmonic progression. (The three lines lie on the same side point A)

3 2. A straight line through the point ( 2, 3) cuts the line x + 3y = 9 and x + y + 1 = 0 at B and C respectively. Find the equation of the line if AB.AC = A line which makes an acute angle with the positive direction of x-axis is drawn through the point P(3, 4) to cut the curve y 2 = 4x at Q and R. Show that the lenghts of the segments PQ and PR are numerical values of the roots of the equation r 2 sin 2 + 4r(2 sin cos) + 4 = A straight lien through A( 15, 10) meets the liens x y 1 = 0, x + 2y = 5 and x + 3y = respectively at A, B and C. If + =, prove that the line passed through the origin. AB AC AD 5. The base AB of a triangle ABC passes through the point (1, 5) which divides in it the ratio 2 : 1. If the equations of the sides AC and BC are 5x y 4 = 0 and 3x 4y 4 = 0 respectively, then find the coordinates of the vertex A. Drill Exercise Find the distance between the line 12 x 5y + 9 = 0 and the point (2, 1). 2. If p is the length of the perpendicular from the origin to the line a x + b y = 1, then prove that = a b 2 p 3. If p and p be the perpendicular from the origin upon the straight lines x sec + y cosec = a and x cos y sin = a cos 2. Prove that 4p 2 + p 2 = a 2. Drill Exercise Find the perpendicular distance of the point (1, 0) from the line 3x + 2y 1 = 0, Also find the co-ordinate of the foot of perpendicular. 2. Find the image of the point (4, 13) in the line 5x + y + 6 = The point P() undergoes a reflection in the x-axis followed by a reflection in the y-axis. Show that their combined effect is the same as the single reflection of P() in the origin when > The image of the point A (1, 2) by the line mirror y = x is the point B and the image of B by the line mirror y = 0 is the point (). Find and. 5. The point P (4, 1) undergoes the following three transformations successively (i) reflection about the line y = x. (ii) translation through a distance 2 units along the positive direction of x-axis. (iii) rotation through an angle /4 about the origin in the anticlockwise direction. Then find the coordinates of the final position.

4 Drill Exercise Find the equation of the straight line that passes through the point (3, 4) and perpendicular to the line 3x + 2y + 5 = Find the equation of a straight line parallel to 2x + 3y + 11 = 0 and which is such that the sum of its intercepts on the axes is Find the equation of the straight line which passes through the point (2, 3) and the point of intersection of the lines x + y + 4 = 0 and 3x y 8 = 0. Drill Exercise Prove that the lines 3x + y 14 = 0, x 2y = 0 and 3x 8y + 4 = 0 are concurrent. 2. Find the value of, if the lines 3x 4y 13 = 0, 8x 11 y 33 = 0 and 2x 3y + = 0 are concurrent. 3. If the lines a 1 x + b 1 y + 1 = 0, a 2 x + b 2 y + 1 = 0 and a 3 x + b 3 y + 1 = 0 are concurrent, show that the points (a 1, b 1 ), (a 2, b 2 ) and (a 3, b 3 ) are collinear. Drill Exercise Find the angles between the pairs of straight lines (i) x y 3 5 = 0 and 3 x + y 7 = 0 (ii) y = (2 3 ) x + 5 and y = (2 + 3 ) x 7 2. Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively. 3. Prove that the straight lines (a + b) x + (a b) y = 2ab, (a b) x + (a + b) y = 2 ab and x + y = 0 form an isosceles triangle whose vertical angle is 2 tan 1 a. b Drill Exercise Find the equation of the bisector of the angles between the straight lines 3x - 4y + 7 = 0 and 12x - 5y - 8 = Find the equation of the obtuse angle bisector of lines 12x - 5y + 7 = 0 and 3y - 4x - 1 = 0.

5 3. Find the bisector of the acute angle between the lines 3x + 4y - 11 = 0 and 12 x - 5y - 2 = 0. Drill Exercise For the straight lines 4x + 3y 6 = 0 and 5x + 12 y + 9 = 0 find the equation of the bisector of the angle which contains the origin. 2. Find the coordinates of the incentre of the triangle whose sides are x + 1 = 0, 3x 4y 5 = 0, 5x + 12y 27 = The sides of a triangle are x y + 3 = 0, 7x y + 3 = 0 and x + y + 1 = 0. Find the equations of the external bisectors of the angles at B and C. Also find the coordinates of the centre of the circle escribed to side BC. Drill Exercise What will be the new coordinates of the point A (1, 2) if origin is shifted to the point at ( 2, 3). 2. At what point the origin be shifted, if the coordinates of a point (4, 5) become ( 3, 9)? 3. If the axes are shifted to the point (1, 2) without rotation, what do the following equations become? (i) 2x 2 + y 2 4x + 4y = 0 (ii) y 2 4x + 4y + 8 = 0 4. Shift the origin to a suitable point so that the equation y 2 + 4y + 8x 2 = 0 will not contain term in y and the constant term. 5. Verify that the area of the triangle with vertices (2, 3), (5, 7) and ( 3, 1) remains invariant under the translation of the axes when the origin is shifted to the point ( 1, 3) Drill Exercise What will be the new coordinates of point A (1, 3) if coordinate axes is rotated by 45º in anticlockwise direction. 2. What was the old coordinates of point A (2, 5) if coordinate axes is rotated by 30º in clockwise direction. 3. If the axes be turned through an angle tan 1 2, what does the equation 4xy 3x 2 = a 2 become? 4. If (x, y) and (X, Y) be the coordinates of the same point referred to two sets of rectangular axes with the same origin and if ax + by become px + qy, where a, b are independent of x, y, prove that a 2 + b 2 = p 2 + q If the axes are shifted to the point ( 2, 3) and then they are rotated through an angle of 45º in anticlockwise sense, what does the equation 2x 2 + 4xy 5y x 22y 24 = 0 become? Drill exercise For what value of does the equation 6x 2 42xy + 60y 2 11x + 10y + = 0 represent two straight lines? 2. Prove that the angle between the straight lines given by,

6 (x cos y sin ) 2 = (x 2 + y 2 ) sin 2 is Show that the difference of the tangents of the angles which the lines, x 2 (sec 2 sin 2 ) 2xy tan + y 2 sin 2 = 0 make with x-axis is Find the equation of the lines bisecting the angles between the pair of lines 3x 2 + xy 2y 2 = If the pairs of straight lines x 2 2pxy y 2 = 0 and x 2 2qxy y 2 = 0 be such that each pair bisects the angle between the other pair, prove that pq = 1. drill exercise Find the equation of the lines joining the origin to the points of intersection of the line x + y = 1 with the curve 4x 2 + 4y 2 + 4x - 2y - 5 = 0, and show that they are at right angles. 2. Find the condition that the pair of straight lines joining the origin to the intersection of the line, y = mx + c and the curve x 2 + y 2 = a 2 may be at right angles. 3. Find the equations of the lines joining the origin to the points of intersection of the curve 2x 2 + 3xy - 4x + 1 = 0 and the line 3x + y = Show that the lines joining the origin to the points of intersection of the line fx gy and the curve x 2 + hxy - y 2 + gx + fy + c = 0 are at right angles for all R if c = The line x my n 0cuts the parabola y 2 = 4ax at P and Q. Find the condition for OP OQ where O is the origin. drill exercise Show that the straight line joining origin to the points of intersection of the curve ax 2 + 2hxy + by gx = 0 and a x 2hxy by 2gx 0 will be at right angles if g(a b ) g(a b). 2. Prove that the angle between the lines joining the origin to the points of intersection of the line y = 3x + 2 with the curve x 2 + 2xy + 3y 2 + 4x + 8y = 11 is 2 2 tan The circle x 2 + y 2 = a 2 cuts off an intercept on the straight line x my 1, which subtends an angle of at the origin. Show that a m 1 a m A pair of straight lines drawn through the origin form an isosceles triangle right angled at the origin with the line 2x + 3y = 6. Find the equation of the pair of straight lines and the area of the triangle. 5. If the lines ax 2 + 2h xy + by 2 = 0 form two adjacent sides of a parallelogram and the line n my 1 is one diagonal, prove that the equation of the other diagonal is, y(b hm) x(am h).

7 ANSWER - KEY Drill exercise (a 2 b ) cos 3. (0, 1) 4 Drill exercise 2 1. (i) 1 : 2 externally (ii) 2 : 5 externally 2. ( b, b) 3. ( 7, 14) Drill exercise 3 1. (0, 0) 2. 11, 2 6 Drill exercise area = 20 sq. unit, length of altitude = k = 3 Drill exercise 5 1. y 2 8y 4x + 20 = 0 2. x 2 + y 2 + 2x 3y 13 = y 2 = x Drill exercise 6 1. (a) x = 2 2. sin = ± 5 3, cos = ± 5 4 Drill exercise 7 1. opposite sides 2. ( 1, 2) Drill exercise x + y = x + 2y + 6 = 0 3. x 5y + 23 = 0, 7x + 4y 8 = 0, 8x y + 15 = 0 Drill exercise 9 1. (2 ± 2 3, 3 ± 2)

8 3. 28x 3y = 49 Drill exercise x - y = 1, 3x - y + 3 = , Drill exercise Drill exercise ,, ( 1, 14) 4. 2, 1 5. (i) reflection of P is (1, 4) 1 7 (iii), 2 2 Drill exercise x 3y + 6 = x + 3y 18 = x y 7 = 0 2. = 7 1. (i) 90º (ii) 60º 2. Drill exercise 14 Drill exercise Drill exercise x + 27 y = 0, 99 x - 77 y + 51 = x + 7y + 11 = x + 3y - 17 = 0 Drill exercise x + 9y 3 = , 3 3

9 3. x + 2y 6 = 0, 2x y + 3= 0, (2, 1) Drill exercise 18 1 (3, 1) 2. (7, 4) 3 3. (i) 2X 2 + Y 2 = 6 (ii) Y 2 = 4X 4., 2 4 drill exercise , , X 2 4Y 2 = a 2 5. X 2 14XY 7Y 2 12 = 0 drill exercise = x 2 10xy y 2 = 0 drill exercise x 2-3y 2-8xy = c 2 = a 2 (1 + m 2 ) 3. x 2 - y 2-5xy = a n y xy 5x 2 = 0, 2.77 sq. units. drill exercise 22

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3 Drill Exercise -. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ).. Prove that the points (a, 4a) (a, 6a) and (a + 3 a, 5a) are the vertices of an equilateral triangle.