Condition for Concurrency of Three Lines

Q. If the diagram in Fig. 2.22 shows the graph of the polynomial f(x) = ax² + bx + c, then

(a) a > 0, b < 0 and c > 0
(b) a < 0, b < 0 and c < 0
(c) a < 0, b > 0 and c > 0
(d) a < 0, b > 0 and c < 0

Q. Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are :

(i) y = x, y = 2x and y + x = 6
(ii) y = x, 3y = x, x + y = 8

Q.

The points A(2, 3), B(-5, 0) and C(-2, a) are collinear , find the value of 'a'.

1. $\frac{9}{7}$

2. $\frac{7}{9}$

3. a = $\frac{13}{7}$

4. a = $\frac{7}{13}$

Q.

At what point do the lines of equations 3x + 2y + 5 = 0 and 2x − 3y − 1 = 0 intersects each other? Find one more point on each of these lines. Prove that these lines are perpendicular.

Q. The points A (4, -2) B(7, 2)C(0, 9)D(-3, 5) from a parallelogram . Find the length of the altitude on the base AB.

Q.

The points (a, b + c), (b, c + a) and (c, a + b) are

1. Non-collinea

2. Collinear

3. Concurrent

4. All of the above

Q.

Plot the points P(3, 2) & Q(-3, -2). From P & Q, draw ⏊s PM & QN on the – ais.

Assign a special name to the geometrical figure PMQN and find its area.

1. Parallelogram, 12

2. Rhombus, 12

3. Square, 12

4. Square, 24

5. Rhombus, 124

6. Parallelogram, 24

Q. Prove that the points (3, 0), (4, 5), (−1, 4) and (−2 −1), taken in order, form a rhombus. Also, find its area.

Q.

If the points A(1, 2) , O(0, 0), C(a, b) are collinear then find ratio a is to b.

Please give the answer based on those three methods of co-ordinate geometry and not on slope and alll

Q.

Determine graphically the vertices of the triangle, the equations of whose sides are given below:

(i) 2y - x = 8, 5y-x = 14 and y - 2x = 1

(ii) y = x, y = 0 and 3x + 3y = 10

Q. 35. If A (-5, 7), B(-4, -5), C(-1, -6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.

Q. If the points p(-3, 9) , Q(a, b) , R(4, -5)are collinear and a+b=1.Find the value of a & b.

Q.

The points (K, 3), (2, -4) and (-k + 1, -2) are collinear. Find K.

Q.

Show that the lines 2x + 5y = 1, x - 3y = 6 and x + 5y + 2 = 0 are concurrent.

Q. Determine whether the following points are collinear.
(1) A(–1, –1), B(0, 1), C(1, 3)

(2) D(–2, –3), E(1, 0), F(2, 1)

(3) L(2, 5), M(3, 3), N(5, 1)

(4) P(2, –5), Q(1, –3), R(–2, 3)

(5) R(1, –4), S(–2, 2), T(–3, 4)

(6) A(–4, 4), $\mathrm{K}\left(–2,\frac{5}{2}\right),$ N (4, –2)

Q. If the points A(2, 3), B(4, k) and C(6, -3) are collinear points, then the value of k is:

1. -3
2. 0
3. 3
4. 12

Q.

Find the coordinates of the foot of the ⏊ dropped from the point P(3 , 1) to the line AB whose equation is 2 – y = 3.

1. (11 , 7)

2. (-11 , -7)

3. (7 , 11)

4. (11/5, 7/5)

5. (-7 , -11)

Q. The value of x, for which the points (x, -1), (2, 1) and (4, 5) lie on a line is

1. 3
2. 2
3. 1
4. 0

Q. For what value of k, do the equation 3x - y + 8 = 0 and 6x - k y = -16 represent coincident lines?

Q. The points (-3, 2), (2, -1) and (a, 4) are collinear. Find a.

1. -19/3
2. -17/3
3. -3/19
4. -5/19

Q.

The x-coordinate of point C is ___.

Q. The length of the perpendicular from the origin to a line is $7$ and the line makes an angle of ${150}^o$ with the positive direction of $y$ axis. The equation of the line is:

1. $\sqrt{3}x+y=14$
2. $\sqrt{3}x-y=13$
3. $\sqrt{3}x+y+14=0$
4. $\sqrt{3}x-y+13=0$

Q.

The coordinates of the point of trisection of the line segment joining the points A(4, 8) and B(-2, 4) are

1. $\begin{array}{c}P=(2,\frac{20}{3})\\ Q=(0,\frac{16}{3})\end{array}$

2. $\begin{array}{c}P=(-4,\frac{8}{3})\\ Q=(10,-9)\end{array}$

3. $\begin{array}{c}P=(8,-9)\\ Q=(4,\frac{-2}{3})\end{array}$

4. $\begin{array}{c}P=(4,9)\\ Q=(-5,\frac{4}{5})\end{array}$

Q. {104 . The circle }x^2+y^2-4x-4y+4=0 is inscribed in a triangle which has two of its sides }} along the coordinate axes. The locus of the circumcentre of the triangle is }{x+y-xy+k(x^2+y^2)^{1/2}=0. Then }k=}{ 2) }1

Q. The points (K, 3), (2, -4) and (-k + 1, -2) are collinear. Find K.

1. -1/3
2. -2/3
3. -1
4. -4

Q. Two vertices of a triangle are the points (1, 4) and (7, 2). Its centroid is the point (5, 3), then the third vertex is :

1. (0, 0)
2. (3, 7)
3. (7, 3)
4. (1, 1)

Q. Find the area of a triangle ABC whose vertices are A(-2 , 2) B(5 , 2) and whose centroid is (1 , 3)

1. 21/2
2. 23/2
3. 19/2
4. 25/2

Q.

Find the slopes of AB & BC given A(2 , 3); B(8 , 11); C(-1 , -1). Also write the relation between points A, B, C.

1. 

2. 

3. 

4. 

5. 

Q.

If the points (, y), (, b) & ( – , y – ) are collinear, find bx = ?

1. ay²

2. ay

3. a + y

4. ay + y

5. a + y²

Q.

The line 4 – 3y + 12 = 0 meets the y-axis at A. Write down the coordinates of A.

Determine the equation of the line passing through A and perpendicular to 4 – 3y + 12 = 0.

1. A(0 , 4); 3 + 4y = 16

2. A(0 , 4); + 4y = 9

3. A(0 , -4); 3 + y = -9

4. A(0 , -4); - 4y = 10

5. A(0 , 2); - 4y = 6