Area of the triangle formed by the points \[\left( (a + 3)(a + 4), a + 3 \right), \left( (a + 2)(a + 3), (a + 2) \right) \text { and } \left( (a + 1)(a + 2), (a + 1) \right)\]

none of these The given points are \[(\left\{ a + 3)(a + 4), \left( a + 3 \right) \right\}, \left\{ (a + 2)(a + 3), (a + 2) \right\} \text { and } \left\{ (a + 1)(a + 2), (a + 1) \right\}\]. Let A be the area of the triangle formed by these points. \[\text { Then, } A = \frac{1}{2}\left[ x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right]\] \[ \Rightarrow A = \frac{1}{2}\left[ \left( a + 3 \right)\left( a + 4 \right)\left( a + 2 - a - 1 \right) + \left( a + 2 \right)\left( a + 3 \right)\left( a + 1 - a - 3 \right) + \left( a + 1 \right)\left( a + 2 \right)\left( a + 3 - a - 2 \right) \right]\] \[ \Rightarrow A = \frac{1}{2}\left[ \left( a + 3 \right)\left( a + 4 \right) - 2\left( a + 2 \right)\left( a + 3 \right) + \left( a + 1 \right)\left( a + 2 \right) \right]\] \[ \Rightarrow A = \frac{1}{2}\left[ a^2 + 7a + 12 - 2 a^2 - 10a - 12 + a^2 + 3a + 2 \right]\] \[ \Rightarrow A = 1\]

Area of the Triangle Formed by the Points ( ( a + 3 ) ( a + 4 ) , a + 3 ) , ( ( a + 2 ) ( a + 3 ) , ( a + 2 ) ) and ( ( a + 1 ) ( a + 2 ) , ( a + 1 ) ) - Mathematics

Question

Area of the triangle formed by the points $((a + 3) (a + 4), a + 3), ((a + 2) (a + 3), (a + 2)) and ((a + 1) (a + 2), (a + 1))$

Options

25a²
5a²
24a²
none of these

MCQ

Solution

none of these

The given points are $({a + 3) (a + 4), (a + 3)}, {(a + 2) (a + 3), (a + 2)} and {(a + 1) (a + 2), (a + 1)}$ .

Let A be the area of the triangle formed by these points.

$Then, A = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]$

$\Rightarrow A = \frac{1}{2} [(a + 3) (a + 4) (a + 2 - a - 1) + (a + 2) (a + 3) (a + 1 - a - 3) + (a + 1) (a + 2) (a + 3 - a - 2)]$

$\Rightarrow A = \frac{1}{2} [(a + 3) (a + 4) - 2 (a + 2) (a + 3) + (a + 1) (a + 2)]$

$\Rightarrow A = \frac{1}{2} [a^{2} + 7 a + 12 - 2 a^{2} - 10 a - 12 + a^{2} + 3 a + 2]$

$\Rightarrow A = 1$

shaalaa.com

Distance of a Point from a Line

Is there an error in this question or solution?

Q 11 Q 10 Q 12

Chapter 23: The straight lines - Exercise 23.21 [Page 133]

APPEARS IN

RD Sharma Mathematics [English] Class 11

Chapter 23 The straight lines
Exercise 23.21 | Q 11 | Page 133

Video TutorialsVIEW ALL [1]

view
Video Tutorials For All Subjects
Distance of a Point from a Line

video tutorial00:00:02

RELATED QUESTIONS

If the lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect each other then find value of k

Find the distance of the point (–1, 1) from the line 12(x + 6) = 5(y – 2).

What are the points on the y-axis whose distance from the line $\frac{x}{3} + \frac{y}{4} = 1$ is 4 units.

Find perpendicular distance from the origin to the line joining the points (cosΘ, sin Θ) and (cosΦ, sin Φ).

Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x– 7y + 5 = 0 and 3x + y = 0.

A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

Prove that the line y − x + 2 = 0 divides the join of points (3, −1) and (8, 9) in the ratio 2 : 3.

Find the equation of the line whose perpendicular distance from the origin is 4 units and the angle which the normal makes with the positive direction of x-axis is 15°.

Find the equation of the straight line at a distance of 3 units from the origin such that the perpendicular from the origin to the line makes an angle tan⁻¹ $(\frac{5}{12})$ with the positive direction of x-axi .

A line a drawn through A (4, −1) parallel to the line 3x − 4y + 1 = 0. Find the coordinates of the two points on this line which are at a distance of 5 units from A.

Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to a line having slope 3/4.

Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to the line x − 2y = 1.

Find the equation of a line perpendicular to the line $\sqrt{3} x - y + 5 = 0$ and at a distance of 3 units from the origin.

Show that the perpendiculars let fall from any point on the straight line 2x + 11y − 5 = 0 upon the two straight lines 24x + 7y = 20 and 4x − 3y − 2 = 0 are equal to each other.

Find the distance of the point of intersection of the lines 2x + 3y = 21 and 3x − 4y + 11 = 0 from the line 8x + 6y + 5 = 0.

What are the points on X-axis whose perpendicular distance from the straight line $\frac{x}{a} + \frac{y}{b} = 1$ is a ?

Find the perpendicular distance from the origin of the perpendicular from the point (1, 2) upon the straight line $x - \sqrt{3} y + 4 = 0 .$

Show that the path of a moving point such that its distances from two lines 3x − 2y = 5 and 3x + 2y = 5 are equal is a straight line.

If sum of perpendicular distances of a variable point P (x, y) from the lines x + y − 5 = 0 and 3x − 2y + 7 = 0 is always 10. Show that P must move on a line.

Determine the distance between the pair of parallel lines:

y = mx + c and y = mx + d

Determine the distance between the pair of parallel lines:

4x + 3y − 11 = 0 and 8x + 6y = 15

Write the value of θ ϵ $(0, \frac{π}{2})$ for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.

L is a variable line such that the algebraic sum of the distances of the points (1, 1), (2, 0) and (0, 2) from the line is equal to zero. The line L will always pass through

The line segment joining the points (−3, −4) and (1, −2) is divided by y-axis in the ratio

The line segment joining the points (1, 2) and (−2, 1) is divided by the line 3x + 4y = 7 in the ratio ______.

Distance between the lines 5x + 3y − 7 = 0 and 15x + 9y + 14 = 0 is

A plane passes through (1, - 2, 1) and is perpendicular to two planes 2x - 2y + z = 0 and x - y + 2z = 4. The distance of the plane from the point (1, 2, 2) is ______.

The shortest distance between the lines

$\bar{r} = (\hat{i} + 2 \hat{j} + \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})$ and

$\bar{r} = (2 \hat{i} - \hat{j} - \hat{k}) + μ (2 \hat{i} + \hat{j} + 2 \hat{k})$ is

If the tangent to the curve y = 3x² - 2x + 1 at a point Pis parallel toy = 4x + 3, the co-ordinates of P are

The distance of the point P(1, – 3) from the line 2y – 3x = 4 is ______.

Find the points on the line x + y = 4 which lie at a unit distance from the line 4x + 3y = 10.

If the sum of the distances of a moving point in a plane from the axes is 1, then find the locus of the point.

The distance between the lines y = mx + c₁ and y = mx + c₂ is ______.

The value of the λ, if the lines (2x + 3y + 4) + λ (6x – y + 12) = 0 are

Column C₁	Column C₂
(a) Parallel to y-axis is	(i) λ = $- \frac{3}{4}$
(b) Perpendicular to 7x + y – 4 = 0 is	(ii) λ = $- \frac{1}{3}$
(c) Passes through (1, 2) is	(iii) λ = $- \frac{17}{41}$
(d) Parallel to x axis is	λ = 3

A straight line passes through the origin O meet the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. Then, the point O divides the segment Q in the ratio:

Area of the Triangle Formed by the Points ( ( a + 3 ) ( a + 4 ) , a + 3 ) , ( ( a + 2 ) ( a + 3 ) , ( a + 2 ) ) and ( ( a + 1 ) ( a + 2 ) , ( a + 1 ) ) - Mathematics

Advertisements

Advertisements

Question

Options

Solution

APPEARS IN

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Select a course

Area of the Triangle Formed by the Points ( ( a + 3 ) ( a + 4 ) , a + 3 ) , ( ( a + 2 ) ( a + 3 ) , ( a + 2 ) ) and ( ( a + 1 ) ( a + 2 ) , ( a + 1 ) ) - Mathematics

Advertisements

Advertisements

Question

Options

SolutionShow Solution

APPEARS IN

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Select a course

Solution