Co-ordinate Geometry

All questions are compulsory.

Group A (1×8=8)

[ Q. No. 1 ] Write the necessary conditions for two straight lines to be parallel.

[ Q. No. 2 ] What is the slope of the line perpendicular to the line y + 5x = 10 ?

[ Q. No. 3 ] Write the equation of a straight line perpendicular to the line ax + by + c = 0.

[ Q. No. 4 ] If the lines 2x + 3y + 5 = 0 and 3x + ky + 5 = 0 are coincident, what is the value of k ?

[ Q. No. 5 ] Define homogeneous equation of second degree with one example.

[ Q. No. 6 ] Under what condition, the pair of straight lines ax² + 2hxy + by² = 0 will be parallel ?

[ Q. No. 7 ] Write the single equation to represent the pair of lines x + a = 0 and x – a = 0.

[ Q. No. 8 ] The single equation of a pair of lines is xy = 0. Write the separate equations of the two lines.

Group B [(2+2)×8=32]

[ Q. No. 9 ]
a) Write the formula to find the angles between two straight lines. Also find the condition of perpendicularity of these lines. [1+1]
b) Using the condition, show that the lines 3y – 2x – 1 = 0 and 3x + 2y + 5 = 0 are perpendicular to each other. [2]

[Q. No. 10]
a) If the straight lines 2x + 3y + 6 = 0 and ax – 5y + 20 = 0 are perpendicular to each other, find the value of a. [2]
b) If two lines ax + by + c = 0 and px + qy + r = 0 are perpendicular to each other, show that : ap + bq = 0. [2]

[ Q. No. 11 ]
a) If the lines 3x + my = 5 and x/2 + y/3 = 1 are parallel to each other, find the value of m. [2]
b) The vertices of a triangle ABC are A(5, 3), B(10, 2) and C(7, 5). Determine the slope and equation of the altitude BD. (2)

[ Q. No. 12 ]
a) Prove that the straight line passing through the points (3, -4) & (-2, 6) and the straight line having equation 2x+y+3=0 are parallel. [2]
b) Find the equation of a straight line passing through the point (2, -3) and perpendicular to the line 5x-4y+19=0. [2]

[ Q. No. 13 ]
a) Find the separate equation of lines represented by the equation 2x²-3xy-2y²=0. Also write the relation between them, if any. [1+1]
b) Prove that the homogenous equation of second degree ax²+2hxy+by² always represents a pair of straight lines passing through origin. [2]

[ Q. No. 14 ]
a) Find the single equation of the pair of lines passing through point (3, -1) and perpendicular to the lines represented by the equation x²-xy-2y²=0. [2]
b) Find the separate equations of lines represented by kx²+3xy-2(k-1)y²=0 if the two lines are perpendicular to each other. [2]

[ Q. No. 15]
a) Obtain the angles between the lines represented by the equation 33x²-44xy+11y²=0. [2]
b) Prove that an acute angle between the straight lines represented by the equation (p²-3q²)x²+8pqxy+(q²-3p²)y² is 60ᣞ. [2]

[ Q. No. 16]
a) Determine the lines represented by the equation 2x²-5xy-3y²+3x+19y-20=0. [2]
b) Show that 2x²-xy-y²+5x+y+2=0 represents two straight lines intersecting at an angle of tanᐨ¹(3).

Group C (4×5=20)

[ Q. No. 17 ] Consider a rhombus ABCD whose opposite vertices are A(2, 4) and C(8, 10). Find :
a) the midpoint of AC. [1]
b) the slope of line AC. [1]
c) the equation of the diagonal BD. [2]

[ Q. No. 18 ] Find the equation of a line AB passing through the point of intersection of the lines 3x+4y-7=0 and 5x-2y-3=0 and perpendicular to the line XY whose equation is 2x+3y-5=0. Also obtain the angles between the lines AB & XY. [3+1]

[ Q. No. 19 ] Prove that the equation of a straight line passing through the point (a cos³∅, a sin³∅) and perpendicular to the line x sec∅ + y cosec∅ = a is x cos∅ – y sin∅ = a cos 2∅. [4]

[ Q. No. 20 ] Obtain the angle between the straight lines represented by the homogenous equation ax² + 2hxy + by² = 0. Also derive the condition of the lines being parallel and perpendicular. [4]

[ Q. No. 21] Find the separate equations of the pair of lines represented by the equation x² – 2xy cosec A + y² = 0. Also, If B be the angle between the two lines, prove that A + B = 90ᣞ. [2+2]

* Best Of Luck *