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Parabola formulas

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Parabola, every formula

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Conic eccentricity
$\frac{S P}{P M} = e$
e=1 parabola, <1 ellipse, >1 hyperbola
Q1MCQConic e
For a parabola, the eccentricity $e$ is:
- A
  $1$
- B
  $0$
- C
  $<1$
- D
  $>1$
Parabola eccentricity
$e = 1$
Q1MCQEccentricity value
Which conic has $e=1$?
- A
  parabola
- B
  ellipse
- C
  hyperbola
- D
  circle
Standard parabola
$y^{2} = 4 a x$
opens right, a>0
Q1MCQStandard form
$y^{2}=8x$ is a parabola with $4a=$:
- A
  $8$
- B
  $4$
- C
  $2$
- D
  $16$
Vertex, focus
$A = (0, 0), S = (a, 0)$
for y²=4ax
Q1MCQFocus
The focus of $y^{2}=8x$ is:
- A
  $(2,0)$
- B
  $(0,2)$
- C
  $(-2,0)$
- D
  $(8,0)$
Directrix & axis
$x = - a (directrix), y = 0 (axis)$
Q1MCQDirectrix
The directrix of $y^{2}=8x$ is:
- A
  $x=-2$
- B
  $x=2$
- C
  $y=-2$
- D
  $x=-8$
Length of latus rectum
$L L^{'} = 4 a$
Ends of latus rectum
$(a, \pm 2 a)$
Q1MCQEnds of LR
The ends of the latus rectum of $y^{2}=4ax$ are:
- A
  $(a,\pm2a)$
- B
  $(\pm2a,a)$
- C
  $(2a,\pm a)$
- D
  $(a,\pm a)$
Focal distance
$S P = x_{1} + a$
for y²=4ax
Parametric point
$(a t^{2}, 2 a t)$
on y²=4ax
Q1MCQParametric point
A parametric point on $y^{2}=4ax$ is:
- A
  $(at^{2},2at)$
- B
  $(2at,at^{2})$
- C
  $(a\cos t,a\sin t)$
- D
  $(at,at^{2})$
Vertex form
$(y - k)^{2} = 4 a (x - h)$
vertex (h,k)
Q1MCQVertex form
$(y-2)^{2}=8(x-1)$ has vertex:
- A
  $(1,2)$
- B
  $(2,1)$
- C
  $(-1,-2)$
- D
  $(0,0)$
Axis parallel to x-axis
$x = a y^{2} + b y + c$
Q1MCQAxis parallel x
A parabola with axis parallel to the x-axis has the form:
- A
  $x=ay^{2}+by+c$
- B
  $y=ax^{2}+bx+c$
- C
  $x^{2}+y^{2}=r^{2}$
- D
  $xy=c^{2}$
Notation S₁, S₁₁
$S_{1} \equiv y y_{1} - 2 a (x + x_{1}), S_{11} \equiv y_{1}^{2} - 4 a x_{1}$
Q1MCQS₁₁
For $y^{2}=4ax$, $S_{11}$ equals:
- A
  $y_1^{2}-4ax_1$
- B
  $x_1^{2}-4ay_1$
- C
  $y_1^{2}+4ax_1$
- D
  $yy_1-2a(x+x_1)$
Position of a point
$S_{11} < 0 inside, = 0 on, > 0 outside$
Q1MCQPosition of point
If $S_{11}<0$ for a point, it lies:
- A
  inside the parabola
- B
  on the parabola
- C
  outside the parabola
- D
  at the focus
Chord joining t₁,t₂
$y (t_{1} + t_{2}) = 2 x + 2 a t_{1} t_{2}$
Q1MCQChord t₁,t₂
The chord joining $t_1,t_2$ on $y^{2}=4ax$ is:
- A
  $y(t_1+t_2)=2x+2at_1t_2$
- B
  $y=t_1t_2 x$
- C
  $yt=x+at^{2}$
- D
  $yy_1=2a(x+x_1)$
Focal chord condition
$t_{1} t_{2} = - 1$
Q1MCQFocal chord condition
For a focal chord with parameters $t_1,t_2$:
- A
  $t_1t_2=-1$
- B
  $t_1t_2=1$
- C
  $t_1+t_2=0$
- D
  $t_1=t_2$
Length of focal chord
$a (t + \frac{1}{t})^{2}$
Q1MCQFocal chord length
The length of a focal chord (parameter $t$) is:
- A
  $a\left(t+\dfrac1t\right)^{2}$
- B
  $a\left(t-\dfrac1t\right)^{2}$
- C
  $4a$
- D
  $2a$
Minimum focal chord (LR)
$least length = 4 a$
Semi-latus rectum (HM)
$\frac{1}{S P} + \frac{1}{S Q} = \frac{2}{l} = \frac{1}{a}$
Q1MCQHM of focal segments
For a focal chord $PSQ$, $\dfrac{1}{SP}+\dfrac{1}{SQ}$ equals:
- A
  $\dfrac1a$
- B
  $\dfrac1{2a}$
- C
  $\dfrac2a$
- D
  $a$
Tangent at (x₁,y₁)
$y y_{1} = 2 a (x + x_{1})$
Q1MCQTangent at point
The tangent to $y^{2}=4ax$ at $(x_1,y_1)$ is:
- A
  $yy_1=2a(x+x_1)$
- B
  $xx_1=2a(y+y_1)$
- C
  $yy_1=a(x+x_1)$
- D
  $y=mx+\dfrac{a}{m}$
Tangent at parameter t
$y t = x + a t^{2}$
Q1MCQTangent at t
The tangent to $y^{2}=4ax$ at parameter $t$ is:
- A
  $yt=x+at^{2}$
- B
  $y=tx+at^{2}$
- C
  $yt=x-at^{2}$
- D
  $y=mx+\dfrac{a}{m}$
Tangent of slope m
$y = m x + \frac{a}{m}$
Q1MCQTangent of slope m
The tangent of slope $m$ to $y^{2}=4ax$ is:
- A
  $y=mx+\dfrac{a}{m}$
- B
  $y=mx+am$
- C
  $y=mx+a$
- D
  $y=mx\pm a\sqrt{1+m^{2}}$
Condition y=mx+c tangent
$c = \frac{a}{m}$
Q1MCQTangency condition
$y=mx+c$ touches $y^{2}=4ax$ iff:
- A
  $c=\dfrac{a}{m}$
- B
  $c=am$
- C
  $c^{2}=a^{2}(1+m^{2})$
- D
  $c=a$
Point of contact (slope form)
$(\frac{a}{m ^{2}}, \frac{2 a}{m})$
Q1MCQPoint of contact
The point of contact of the slope-$m$ tangent to $y^{2}=4ax$ is:
- A
  $\left(\dfrac{a}{m^{2}},\dfrac{2a}{m}\right)$
- B
  $\left(\dfrac{2a}{m},\dfrac{a}{m^{2}}\right)$
- C
  $(am^{2},2am)$
- D
  $(am^{2},-2am)$
Normal at (x₁,y₁)
$y - y_{1} = - \frac{y _{1}}{2 a} (x - x_{1})$
Q1MCQNormal at point
The normal to $y^{2}=4ax$ at $(x_1,y_1)$ has slope:
- A
  $-\dfrac{y_1}{2a}$
- B
  $\dfrac{y_1}{2a}$
- C
  $\dfrac{2a}{y_1}$
- D
  $-\dfrac{2a}{y_1}$
Normal at parameter t
$y + x t = 2 a t + a t^{3}$
Q1MCQNormal at t
The normal to $y^{2}=4ax$ at parameter $t$ is:
- A
  $y+xt=2at+at^{3}$
- B
  $yt=x+at^{2}$
- C
  $y=mx-2am-am^{3}$
- D
  $y-xt=2at$
Normal of slope m
$y = m x - 2 am - a m^{3}$
Q1MCQNormal of slope m
The normal of slope $m$ to $y^{2}=4ax$ is:
- A
  $y=mx-2am-am^{3}$
- B
  $y=mx+\dfrac{a}{m}$
- C
  $y=mx+2am+am^{3}$
- D
  $y=mx-am$
Foot of normal (slope m)
$(a m^{2}, - 2 am)$
Q1MCQFoot of normal
The foot of the slope-$m$ normal to $y^{2}=4ax$ is:
- A
  $(am^{2},-2am)$
- B
  $(am^{2},2am)$
- C
  $\left(\dfrac{a}{m^{2}},\dfrac{2a}{m}\right)$
- D
  $(a,2a)$
Perpendicular tangents locus
$x + a = 0 (directrix)$
Q1MCQPerpendicular tangents
The locus of intersection of perpendicular tangents to $y^{2}=4ax$ is:
- A
  $x+a=0$ (the directrix)
- B
  the axis
- C
  $x=a$
- D
  the latus rectum
Chord of contact
$y y_{1} = 2 a (x + x_{1}) (S_{1} = 0)$
Q1MCQChord of contact
The chord of contact of $(x_1,y_1)$ w.r.t. $y^{2}=4ax$ is:
- A
  $yy_1=2a(x+x_1)$
- B
  $xx_1=2a(y+y_1)$
- C
  $S_{11}=0$
- D
  $y=mx+\dfrac{a}{m}$
Tangents from external point
$m^{2} x_{1} - m y_{1} + a = 0$
slopes m1,m2
Q1MCQTangent slopes
The slopes of tangents from $(x_1,y_1)$ to $y^{2}=4ax$ satisfy $m^{2}x_1-my_1+a=0$, so $m_1m_2$ is:
- A
  $\dfrac{a}{x_1}$
- B
  $\dfrac{y_1}{x_1}$
- C
  $\dfrac{a}{y_1}$
- D
  $x_1$
Sum/product of tangent slopes
$m_{1} + m_{2} = \frac{y _{1}}{x _{1}}, m_{1} m_{2} = \frac{a}{x _{1}}$
Q1MCQSum of slopes
For tangents from $(x_1,y_1)$ to $y^{2}=4ax$, $m_1+m_2$ equals:
- A
  $\dfrac{y_1}{x_1}$
- B
  $\dfrac{a}{x_1}$
- C
  $\dfrac{x_1}{y_1}$
- D
  $y_1$
Chord with midpoint
$S_{1} = S_{11} : y y_{1} - 2 a (x + x_{1}) = y_{1}^{2} - 2 a x_{1}$
Q1MCQMidpoint chord
The chord of $y^{2}=4ax$ with midpoint $(x_1,y_1)$ is:
- A
  $S_1=S_{11}$
- B
  $S_1=0$
- C
  $S_{11}=0$
- D
  $S=0$
Reflection property
$ray parallel to axis reflects through the focus$
Q1MCQReflection
A ray parallel to the axis of a parabola, after reflection, passes through:
- A
  the focus
- B
  the vertex
- C
  the directrix
- D
  infinity
Angle between tangents
$tan θ = \frac{S _{11}}{x _{1} + a}$
Q1MCQAngle between tangents
The angle between the pair of tangents from $(x_1,y_1)$ to $y^{2}=4ax$ satisfies:
- A
  $\tan\theta=\dfrac{\sqrt{S_{11}}}{x_1+a}$
- B
  $\tan\theta=\dfrac{x_1+a}{\sqrt{S_{11}}}$
- C
  $\tan\theta=S_{11}$
- D
  $\tan\theta=\dfrac{a}{x_1}$
Normal chord: t₂ from t₁
$t_{2} = - t_{1} - \frac{2}{t _{1}}$
Q1MCQNormal chord
If the normal at $t_1$ meets $y^{2}=4ax$ again at $t_2$, then:
- A
  $t_2=-t_1-\dfrac{2}{t_1}$
- B
  $t_2=t_1+\dfrac{2}{t_1}$
- C
  $t_1t_2=-1$
- D
  $t_2=-\dfrac{1}{t_1}$
Circle on focal chord
$touches the directrix$
Q1MCQCircle on focal chord
A circle drawn on a focal chord of a parabola as diameter:
- A
  touches the directrix
- B
  passes through the vertex
- C
  touches the axis
- D
  passes through the focus twice

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Parabola formulas

Parabola, every formula

Conic eccentricity

Parabola eccentricity

Standard parabola

Vertex, focus

Directrix & axis

Length of latus rectum

Ends of latus rectum

Focal distance

Parametric point

Vertex form

Axis parallel to x-axis

Notation S₁, S₁₁

Position of a point

Chord joining t₁,t₂

Focal chord condition

Length of focal chord

Minimum focal chord (LR)

Semi-latus rectum (HM)

Tangent at (x₁,y₁)

Tangent at parameter t

Tangent of slope m

Condition y=mx+c tangent

Point of contact (slope form)

Normal at (x₁,y₁)

Normal at parameter t

Normal of slope m

Foot of normal (slope m)

Perpendicular tangents locus

Chord of contact

Tangents from external point

Sum/product of tangent slopes

Chord with midpoint

Reflection property

Angle between tangents

Normal chord: t₂ from t₁

Circle on focal chord

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