Tag: introduction to slope

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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

A circle of radius 2 is concentric with ellipse $\frac{x^{2}}{7}+\frac{y^{2}}{3}=1$ then inclination of common tangent with x-axis - 

  1. $\frac{\pi }{2}$
  2. $\frac{\pi }{4}$
  3. $\frac{\pi }{3}$
  4. $\frac{\pi }{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For an ellipse x^2/a^2 + y^2/b^2 = 1, the tangent with slope m is y = mx +/- sqrt(a^2m^2 + b^2). For a concentric circle x^2 + y^2 = r^2, the tangent is y = mx +/- r*sqrt(1+m^2). Equating these leads to the vertical tangent case.

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 with negative slope passing the point (1, 4) meets the coordinate axes at A and B. The minimum value of OA + OB = 

  1. 5

  2. 6

  3. 9

  4. 8
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the line pass through (1, 4) with intercepts a and b on the x and y axes, so its equation is x/a + y/b = 1. Substituting the point gives 1/a + 4/b = 1, and minimizing a + b using Cauchy-Schwarz or derivatives yields the minimum value of 9, making option C correct.

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 straight line in space is equally inclined to the co-ordinate axes, the cosine of its angle of inclination to any of the axes is 

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

If a line makes equal angles alpha with all three coordinate axes, then the direction cosines satisfy cos^2(alpha) + cos^2(alpha) + cos^2(alpha) = 1. This simplifies to 3 cos^2(alpha) = 1, meaning cos(alpha) = 1 / sqrt(3), so option C is correct.

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 line equally inclined to the coordinate axes and equidistant from points 
$A(1,-2)and B(3,4) is$

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

Lines equally inclined to axes have slopes 1 or -1. Equidistant from (1, -2) and (3, 4) means they pass through the midpoint (2, 1) or are parallel to the line segment AB. Calculating these yields the lines x+y=3 and x-y=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

the inclination of the tangent at$\theta =\frac { \pi  }{ 3 } on\quad the\quad curve\quad x=a\left( \theta +sin\theta  \right) ,y=a\left( 1+cos\theta  \right) is$

  1. $\frac { \pi }{ 3 }$
  2. $\frac { \pi }{ 6 }$
  3. $\frac { 2\pi }{ 3 }$
  4. $\frac { 5\pi }{ 3 }$
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

The side of a triangle are a. b and $\sqrt{a^2+ab+b^2}$. The greatest angle is 

  1. $90^0$
  2. $135^0$
  3. $120^0$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The greatest angle is opposite the longest side, which is sqrt(a^2 + ab + b^2). Applying the law of cosines with sides a, b, and c = sqrt(a^2 + ab + b^2) gives cos(theta) = (a^2 + b^2 - c^2) / (2ab) = -1/2, meaning theta = 120 degrees.

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 the line passing through origin and making an angle $30^{\circ}$ with xaxis is

  1. $ x=\sqrt 3y$
  2. $ y=\sqrt 3x $
  3. $ x=3y$
  4. $ y=3x $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle made by line is $30^{\circ}$


The slope of line is $m=\tan 30=\dfrac 1{\sqrt 3}$


Equation of line is $y=mx+c$

$y=\dfrac 1{\sqrt 3} x+c$

Put $(x,y)=(0,0)\Rightarrow c=0$

$\Rightarrow y=\dfrac 1{\sqrt 3} x$

$\therefore\ x=\sqrt 3 y$