Tag: introduction to straight lines

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Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

The equation of line passing through $(5,4)$ and $(-9,3)$

  1. $x-14y+51=0 $
  2. $ x+y=2 $
  3. $ x-2 y=0 $
  4. $ 2 x-y=0 $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The equation of line Passing through $(5,4)$ and $(-9,3)$

The slope of line is $\dfrac{3-4}{5+9}=\dfrac 1{14}$

The equation of line is $y-4=\dfrac 1{14}(x-5)$

$\Rightarrow 14y-56=x-5$

$\Rightarrow x-14y+51=0$
Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

The intercepts made by a line on the co-ordinate axes are in the ratio $3:4$

and passes through $(3,0)$

  1. $3x+4y=12$
  2. $4x+3y=12$
  3. $4x-3y+12=0$

  4. None.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the common factor be $k$ 


The ratio is $3:4$


The intersepts are $3k,4k$

The equation is 
$\dfrac x{3k}+\dfrac y{4k}=1\\4kx+3ky=12k^2\\4x+3y=12k$

It passes through $(3,0)$

$\implies 4(3)+3(0)=12k\\k=1$

So the equation is $4x+3y=12$

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

If the coordinates of $\left(x,y\right)$ lie in first quadrant then 

  1. $x = 1, y = 7$
  2. $x = 6, y =-2$
  3. $x = -3, y=7$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In first quadrant $x$ and $y$ are positive. Hence $x$ takes the value $1$ and $y$ takes the value $7$

$\therefore \,x=1,\,y=7$

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

A straight line through the origin 'O' meets the parallel lines 4x+2y=9 and 2x+y+6=0 at points p and q respectively. Then the points o divides the segment PQ in the ratio

  1. 1:2

  2. 3:4

  3. 2:1

  4. 4:3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The origin O divides the segment PQ in the ratio of the distances from the origin to the lines. Using the perpendicular distance formula from (0,0) to the lines 4x+2y-9=0 and 2x+y+6=0, the ratio is 9/12 = 3/4, but the correct ratio for the segment division is 1:2.

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

Normals are drawn at points $A, B, and C$ on the parabola ${ y }^{ 2 }= { 4x }$ which intersect at P\left( h, k \right)$. The locus of the point $P$ if the slope of the line joining the feet of two of them is $2$, is

  1. ${ x + y = 1 }$
  2. ${ x - y = 3 }$
  3. ${ { y }^{ 2 } = 2\left(x-1\right) }$
  4. ${ { y }^{ 2 }=2\left( x-\frac { 1 }{ 2 } \right) }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

Find the slope of $\displaystyle x\cot  \alpha -y\tan \alpha =1$

  1. $\displaystyle cosec^{2}\alpha -1$
  2. $\displaystyle \sec ^{2}\alpha -1$
  3. $\displaystyle \tan ^{2}\alpha -1$
  4. $\displaystyle \cot ^{2}\alpha -1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, $ x \cot \alpha - y \tan \alpha = 1 $
$ =>  y \tan \alpha = x \cot \alpha - 1 $
$ => y = x \dfrac {\cot \alpha}{\tan \alpha}  - \dfrac {1}{\tan \alpha} $

The slope of the equation of line  of the form $ y = mx + c $ is $ m $
So, slope of the given line $ = \dfrac {\cot \alpha}{\tan \alpha} = \cot ^ 2 (\alpha) = cosec^2 (\alpha) - 1 $

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

If a line passes through the point $P(1, 2)$ makes an angle of $45^o$ with the x-axis and meets the line $x+2y-7=0$ in Q, then PQ equals?

  1. $\dfrac{2\sqrt{2}}{3}$
  2. $\dfrac{3\sqrt{2}}{2}$
  3. $\sqrt{3}$
  4. $\sqrt{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The line through (1, 2) with angle 45 degrees is y - 2 = 1(x - 1), or y = x + 1. Intersection with x + 2y - 7 = 0: x + 2(x + 1) - 7 = 0 => 3x = 5 => x = 5/3, y = 8/3. Distance PQ = sqrt((5/3 - 1)^2 + (8/3 - 2)^2) = sqrt((2/3)^2 + (2/3)^2) = sqrt(8/9) = 2*sqrt(2)/3.

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

Find the equation of a line which makes equal angles with the lines $x+y-2=0$ and $7x-y+4=0$ and passes through $(1, 2)$.

  1. $x-3y=5$
  2. $3x+y=5$
  3. $7x-y=5$
  4. $x+3y=7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle bisectors of the lines x+y-2=0 and 7x-y+4=0 are found using the formula (x+y-2)/sqrt(2) = +/- (7x-y+4)/sqrt(50). Simplifying these and checking which passes through (1, 2) gives the equation.

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

State the following statement is true(T) or false(F).
If two lines intersect at a point P, then P is called the point of concurrent of the two lines.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For $2$ lines

      We call it as Intersection point
For $3$ lines
      We call it as point of concurrent

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

If $O$ is origin and $C$ is the mid point of $A(2,-1)$ and $B(-4,3)$. Then value of $\overrightarrow { OC } $ is

  1. $\hat { i } +\hat { j } $
  2. $\hat { i } -\hat { j } $
  3. $-\hat { i } +\hat { j } $
  4. $-\hat { i } -\hat { j } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since $C$ is the mid point of $A(2,-1)$ and $B(-4,3)$


$\therefore$ Coordinates of $C$ are
$\left( \cfrac { 2-4 }{ 2 } ,\cfrac { -1+3 }{ 2 }  \right) =\left( -1,1 \right) $
$\therefore \overrightarrow { OC } =-\hat { i } +\hat { j } $