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STRAIGHT LINES
1.The points A(2,-2), B(3,-5) and C(5,7) are the vertices of a triangle. Find the equation
of the perpendicular from `C on to the internal bisector of angle `A
2.Two straight lines including an angle θ cut off segments from co-ordinate axes,each
equal to `k . Show that the locus of their point of intersection is x± y± k cotθ =0.
3 A triangle has (2,6) as a vertex and x-7y+15=0 and 7x+y+5=0 as an altitude and
angular bisector respectively drawn from one and the same vertex. Find the circumcentre
of the triangle.
4. A triangle ABC, right angled at C moves such that the points A and B always lie on the
co-ordinate axes and C and origin 0 are not on the same side if AB= 13 and BC =5. Find
the locus of C.
5.One side of a square is inclined to the x-axis at an angle α and one of its extremities is
at the origin. If a is the side of the square, prove that x(cosα -sinα ) +y (sinα +cosα)=a
is one of its diagonal.
6.Given A(1,2), B(5,3) are two vertices of a triangle. Find the third vertex of the triangle
if it lies on x+2y-2=0 and the perimeter of the triangle is least.
7.In ∆ABC A = (1,2) orthocentre is (4,2) .If AB = 5 find the coordinates of B and C .
8.The ∆ABC is right angled at C. If A= (1,2) ,AC=8 and circumcentre is (6,2) find C.
9.In ∆ABC B=60 and C=90o. Equations to BC is 5x-12y-6=0 . If incenter is (1,1). Find
the lengths of the sides.
10.Find the locus of the image of any point on the curve 2x2+y2=1 with respect to the line
2x+y-2=0.
11.Find the equation of a line through the intersection of 3x+4y+6=0, x+y+2=0 and
farthest from (2,3) .
12.Two consecutive sides of a parallelogram are 4x+5y=0,7x+2y=0 . If 11x+7y=9 is a
diagonal find the equation of other sides and diagonal without finding the vertices.
13.The base of a right angled triangle is 2x-y-3=0 and its centroid and circumcentre are
(4,3) and (5,4) respectively. Find the orthocentre and area of the triangle.
14.In ∆ABC equations of AB and AC are x+y=2 and x-2y=1 respectively. If P (-2,1) lies
on BC such that 1/BP + 1/CP is maximum find the equation to BC.
15.If a,b,c are in GP and b3 ,a3, c3 are in AP show that the lines ax+by+c=0,bx+cy+a=0
and cx+ay+b=0 are concurrent.
16.Write the equations of the sides of a triangle knowing one of its vertices B(2,-7) and
the equation of the altitude 3x+y+11=0 and median x+2y+7=0 drawn from different
vertices.
17.Write the equation of the straight line parallel to the two lines 3x-2y-1=0 (x1)/2=(y+5)/3 and passing in the middle between them.
18.All possible straight lines are drawn through the point P(-3,-1). Prove that the lines x2y-3=0 and x-2y+5=0, cut from each of them a segment whose mid-point is at P.
19.'P' is the point (-1,2). A variable line through P cuts the co-ordinate axes at A and B.
Q is a point on AB such that PA,PQ,PB are in H.P. Show that locus of Q is 2x-y=0.
20.The equation (5x+2y+4)+k(x+9y-25)=0 represents a set of concurrent lines. Find the
equations of those lines belonging to this set which together with the lines 2x-3y+5=0
and 12x+8y-7=0 from isoceles triangle.
21.The straight line x+y=1 meets the co-ordinate axes at P and Q and another line
perpendicular to it meets the axes at R and S. Find the locus of intersection of the lines
PS and QR.
22.Find the equation of the straight line belonging to both the systems a(2x-3y-10) +
b(x+2y-6)=0 and c(16x-10y-33) + d(12x-14y+29)=0.
23.The straight line ax+by+c=0 cuts the locus of intersection of the lines (tx/4)-(y/3)+t=0
and (x/4)+(ty/3)-1=0 at A and B such that line AB subtends a right angle at the origin,
find the distance of the given line from the origin.
24.If the lines ti x + y = 2ati + a ti 3 where i = 1,2,3 are concurrent find 1/t1t2 + 1/t2t3 +
1/t3t1.
25.Find the range of a for which the point (cosα ,sinα ) lies inside the ∆ formed by
x+y=2, x-y=1 and 6x+2y=√10.
ANSWERS OF STRAIGHT LINES
1.x=5
3.(5/2,5/2)
4.st.line
6. (11/10,9/20)
8. (37/5,34/5) or (37/5-24/5)
9.√3+1, √3+3, 2√3+2
10. 2(3x+4y-8)2 + (4x-3y-4)2 = 25
11. 4x+3y+8=0
12. 4x+5y-9=0; 7x+2y=9,x=y
13. (2,1),54/5
14.11x-2y+24=0
16. 4x+3y+13=0; 7x+9y+19=0; x-3y=23
17. 3x-2y=7 20; x+5y=13; 5x-y+13=0
21. x2 +y2-x-y=0
22. 5x-2y=7
23. 12/5
24. 0
25. 0< α <5π /6 - arctan 3