Question:
The locus of the vertices of the family of parabola $ 6y = 2a^3x^2 + 3a^2x - 12a $ is
The locus of the vertices of the family of parabola $ 6y = 2a^3x^2 + 3a^2x - 12a $ is
Updated On: Jul 7, 2022
- $xy=\frac{105}{64}$
- $xy=\frac{64}{105}$
- $xy=\frac{35}{16}$
- $xy=\frac{16}{35}$
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The Correct Option is A
Solution and Explanation
Given, $6y=2a^{3}x^{2}+3a^{2}x-12a\quad...\left(1\right)$
For a vertex of given equation, $\frac{dy}{dx} = 0 $
$ \therefore 6 \frac{dy}{dx} = 4a^{3}x+3a^{2}= 0 $
$\Rightarrow a= -\frac{3}{4x}$
Putting the value of a in $\left(1\right)$, we get
$ 6y=2\left(\frac{-3}{4x}\right)x^{2} + 3\left(-\frac{3}{4x}\right)^{2} -12\left(-\frac{3}{4x}\right)$ $\Rightarrow6y= -\frac{27}{32x}+\frac{27}{16x}+\frac{36}{4x} $
$ \Rightarrow 192xy = -27+54+288 $
$\Rightarrow xy= \frac{315}{192} $
$= \frac{105}{64}$
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Top Questions on Parabola
- If the line \( 3x - 2y + 12 = 0 \) intersects the parabola \( 4y = 3x^2 \) at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to:
- Let A and B be the two points of intersection of the line \( y + 5 = 0 \) and the mirror image of the parabola \( y^2 = 4x \) with respect to the line \( x + y + 4 = 0 \). If \( d \) denotes the distance between A and B, and \( a \) denotes the area of \( \Delta SAB \), where \( S \) is the focus of the parabola \( y^2 = 4x \), then the value of \( (a + d) \) is:
- Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:
- Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:
- Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at points A and B, then \( (AB)^2 \) is equal to:
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Notes on Parabola
Concepts Used:
Parabola
Parabola is defined as the locus of points equidistant from a fixed point (called focus) and a fixed-line (called directrix).
Standard Equation of a Parabola
For horizontal parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A,
- Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,
- P(x,y) as the moving point.
- Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola.
- The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ.
- The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure.
- By definition PM = PS
=> MP2 = PS2
- So, (a + x)2 = (x - a)2 + y2.
- Hence, we can get the equation of horizontal parabola as y2 = 4ax.
For vertical parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A
- Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
- P(x,y) as any moving point
- Let us now draw a perpendicular SZ from S to the directrix.
- Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ.
- The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
- By definition PM = PS
=> MP2 = PS2
So, (b + y)2 = (y - b)2 + x2
- As a result, the vertical parabola equation is x2= 4by.