Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a. - Mathematics

Advertisements
Advertisements
Sum

Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.

Advertisements

Solution

Given, the points are (a + 5, a – 4), (a – 2, a + 3) and (a, a).

We have to prove that these pints do not lie on a straaighine.

So, we have to prove that these points form a triangle.

Area, Δ = `1/2|("a" + 5, "a" - 4, 1),("a" - 2, "a" + 3, 1),("a", "a", 1)|`

[Applying R1 → R1 – R3 and R2 → R2 – R3]

= `1/2 |(5, -4, 0),(-2, 3, 0),("a", "a", 1)|`

= `1/2[(1 * (15 - 8)]`

= `7/2 ≠ 0`

Hence, given points from a triangle i.e., points do not lie on a straight line.

  Is there an error in this question or solution?
Chapter 4: Determinants - Exercise [Page 78]

APPEARS IN

NCERT Exemplar Mathematics Class 12
Chapter 4 Determinants
Exercise | Q 15 | Page 78

RELATED QUESTIONS

Find the area of a triangle whose vertices are A(3, 2), B (11, 8) and C(8, 12).


The two opposite vertices of a square are (− 1, 2) and (3, 2). Find the coordinates of the other two vertices.


Find the area of a triangle with vertices at the point given in the following:

(2, 7), (1, 1), (10, 8)


Find the area of a triangle with vertices at the point given in the following:

(−2, −3), (3, 2), (−1, −8)


Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.


Find values of k if area of triangle is 4 square units and vertices are (k, 0), (4, 0), (0, 2)


Find values of k if area of triangle is 4 square units and vertices are (−2, 0), (0, 4), (0, k)


Find equation of line joining (1, 2) and (3, 6) using the determinant.


Find equation of line joining (3, 1) and (9, 3) using determinant.


If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4), then k is ______.


Find the area of the following triangle:


Find the area of the following triangle:


Find the missing value:

Base Height Area of triangle
15 cm ______ 87 cm2

ΔABC is isosceles with AB = AC = 7.5 cm and BC = 9 cm (see the given figure). The height AD from A to BC, is 6 cm. Find the area of ΔABC. What will be the height from C to AB i.e., CE?


Find the area of a triangle whose vertices are

(6,3), (-3,5) and (4,2)


Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm ?


Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm ?


Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42cm ?


The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the area of the triangle ?


Find the area of the blades of thc magnetic compass shown in Fig.. 12.27. (Take √11 = 3.32).


Find the centroid of  ΔABC  whose vertices are A(-1, 0) B(5, -2) and C(8,2) 

 


Find the area of  ΔABC with A(1, -4) and midpoints of sides through A being (2, -1) and (0, -1).


For what value of k(k>0) is the area of the triangle with vertices (-2, 5), (k, -4) and (2k+1, 10) equal to 53 square units?


Find a relation between x and y, if the points A(x, y), B(-5, 7) and C(-4, 5) are collinear.


If the area of triangle ABC formed by A(x, y), B(1, 2) and C(2, 1) is 6 square units, then prove that x + y = 15 ?


The coordinates of the point P dividing the line segment joining the points A (1, 3) and B (4, 6) in the ratio 2 : 1 are:


 Using determinants, find the values of k, if the area of triangle with vertices (–2, 0), (0, 4) and (0, k) is 4 square units. 


In ☐ ABCD, l(AB) = 13 cm, l(DC) = 9 cm, l(AD) = 8 cm, find the area of ☐ ABCD.


What is the area of a triangle with base 4.8 cm and height 3.6 cm?


Find the area of the following triangle:


Find the area of the following triangle:


If the sides of a triangle are 3 cm, 4 cm and 5 cm, then the area is 


The table given below contains some measures of the right angled triangle. Find the unknown values.

Base Height Area
20 cm 40 cm ?

The table given below contains some measures of the right angled triangle. Find the unknown values.

Base Height Area
5 feet ? 20 sq.feet

The table given below contains some measures of the right angled triangle. Find the unknown values.

Base Height Area
? 12 m 24 sq.m

A field is in the shape of a right angled triangle whose base is 25 m and height 20 m. Find the cost of levelling the field at the rate of ₹ 45 per sq.m2


If Δ = `|(1, x, x^2),(1, y, y^2),(1, z, z^2)|`, Δ1 = `|(1, 1, 1),(yz, zx, xy),(x, y, z)|`, then prove that ∆ + ∆1 = 0.


In a triangle ABC, if `|(1, 1, 1),(1 + sin"A", 1 + sin"B", 1 + sin"C"),(sin"A" + sin^2"A", sin"B" + sin^2"B", sin"C" + sin^2"C")|` = 0, then prove that ∆ABC is an isoceles triangle.


If A, B, C are the angles of a triangle, then ∆ = `|(sin^2"A", cot"A", 1),(sin^2"B", cot"B", 1),(sin^2"C", cot"C", 1)|` = ______.


If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x1, y1), (x2, y2), (x3, y3), then `|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4`


Show that the ∆ABC is an isosceles triangle if the determinant

Δ = `[(1, 1, 1),(1 + cos"A", 1 + cos"B", 1 + cos"C"),(cos^2"A" + cos"A", cos^2"B" + cos"B", cos^2"C" + cos"C")]` = 0


The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq.units. The value of k will be ______.


Find the area of the triangle whose vertices are (-2, 6), (3, -6), and (1, 5).


Let `Delta = abs (("x", "y", "z"),("x"^2, "y"^2, "z"^2),("x"^3, "y"^3, "z"^3)),` then the value of `Delta` is ____________.


The points (1,1), (-2, 7) and (3, -3) are ______.


The area of a triangle with base 4 cm and height 6 cm is 24 cm2.


The area of ∆ABC is 8 cm2 in which AB = AC = 4 cm and ∠A = 90º.


The base and the corresponding altitude of a parallelogram are 10 cm and 3.5 cm, respectively. The area of the parallelogram is 30 cm2.


Find the cost of laying grass in a triangular field of sides 50 m, 65 m and 65 m at the rate of Rs 7 per m2.


The dimensions of a rectangle ABCD are 51 cm × 25 cm. A trapezium PQCD with its parallel sides QC and PD in the ratio 9:8, is cut off from the rectangle as shown in the figure. If the area of the trapezium PQCD is `5/6` h part of the area of the rectangle, find the lengths QC and PD.


The area of a triangle with vertices A, B, C is given by ______. 


Ratio of areas of ∆MNO, ∆MOP and ∆MPQ in the given figure is ______.


In the given figure, ΔMNO is a right-angled triangle. Its legs are 6 cm and 8 cm long. Length of perpendicular NP on the side MO is ______.


Area of a triangle = `1/2` base × ______.


Observe all the four triangles FAB, EAB, DAB and CAB as shown in the given figure. 

  1. All triangles have the same base and the same altitude.
  2. All triangles are congruent.
  3. All triangles are equal in area.
  4. All triangles may not have the same perimeter.

Let a vector `αhati + βhatj` be obtained by rotating the vector `sqrt(3)hati + hatj` by an angle 45° about the origin in counter-clockwise direction in the first quadrant. Then the area of triangle having vertices (α, β), (0, β) and (0, 0) is equal to ______.


If (a, b), (c, d) and (e, f) are the vertices of ΔABC and Δ denotes the area of ΔABC, then `|(a, c, e),(b, d, f),(1, 1, 1)|^2` is equal to ______.


Using determinants, find the area of ΔPQR with vertices P(3, 1), Q(9, 3) and R(5, 7). Also, find the equation of line PQ using determinants.


Share
Notifications



      Forgot password?
Use app×