Tag: tracing of the parabola

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Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

The angle of intersection at the origin to the curves ${ y }^{ 2 }=4x$ and ${ x }^{ 2 }=4y$ is :

  1. $\pi $
  2. $\dfrac{ \pi }{ 3}$
  3. $\dfrac{ \pi }{ 6 }$
  4. $\dfrac{ \pi }{ 2 }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given  $y^2=4x$

Differentiating w.r.t. $x$, we get,

$2y\dfrac{dy}{dx}=4$

$\dfrac{dy}{dx}=\dfrac2y.......1$

$x^2=4y$

Differentiating w.r.t. $x$, we get,

$2x=4\dfrac{dy}{dx}$

$\dfrac{dy}{dx}=\dfrac{x}{2}...........2$

So the slope of tangent at $(0,0)$ of $(1)$ is Parallel to $y$ axis 

And the slope of tangent at $(0,0)$ of $(2)$ is Parallel to $x$ axis 

So the angle between them is $\dfrac{\pi}{2}$ 
Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

The parametric equation of a parabola is $x=t^{2}+1, y=2t+1$. The Cartesian equation of its directrix is 

  1. $x=0$
  2. $x+1=0$
  3. $y=0$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given x = t^2 + 1 and y = 2t + 1, we have t = (y - 1)/2. Substituting into x gives x = ((y - 1)/2)^2 + 1, or (y - 1)^2 = 4(x - 1). This is a parabola with vertex (1, 1), opening right, with 4a = 4, so a = 1. The directrix is x = h - a, which is x = 1 - 1 = 0.

Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

$TP$ and $TQ$ are tangents to parabola $y^{2}=4x$ and normal at $P$ and $Q$ intersect at a point $R$ on the curve. The locus of the center of the circle circumscribing $\Delta TPQ$ is parabola whose

  1. Vertex is $\left(1,0\right)$.
  2. Foot of directrix is $\left(\dfrac{7}{8},0\right)$
  3. Length of latus-rectum is $\dfrac{1}{4}$.
  4. Focus is $\left(\dfrac{9}{8},0\right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For the parabola y^2 = 4x, the locus of the circumcenter of triangle TPQ, where P and Q are points on the parabola and the normals at P and Q meet on the parabola, is a known property. The resulting locus is another parabola with vertex (1, 0).