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Locus of the centre of the circle passing through the vertex and the mid-points of perpendicular chords from the vertex of the parabola $y^{2} = 4 a x$ is.

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Important Questions on Parabola

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-2>Q 35

The equation

3 x^{2} + 4 y^{2} - 18 x + 16 y + 43 = c

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-3>Q 1

The tangent

P T

and the normal

P N

to the parabola

y^{2} = 4 a x

at a point

P

on it meet its axis at points

T

and

N

respectively. The locus of the centroid of triangle

P T N

is a parabola whose

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-3>Q 11

Answer $Q . 39, Q . 40$ and $Q . 41$ by appropriately matching the information given in the three columns of the following table.


Column- $1$	Column- $2$	Column- $3$
$(I) x^{2} + y^{2} = a^{2}$	$(i) m y = m^{2} x + a$	$(P) (\frac{a}{m^{2}}, \frac{2 a}{m})$
$(I I) x^{2} + a^{2} y^{2} = a^{2}$	$(i i) y = m x + a \sqrt{m^{2} + 1}$	$(Q) (\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
$(I I I) y^{2} = 4 a x$	$(i i i) y = m x + a \sqrt{m^{2} - 1}$	$(R) (\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
$(I V) x^{2} - a^{2} y^{2} = a^{2}$	$(i v) y = m x + \sqrt{a^{2} m^{2} + 1}$	$(S) (\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

If a tangent to a suitable conic (Column $1$ ) is found to be $y = x + 8$ and its point of contact is $(8, 16),$ then which of the following options is the only CORRECT combination?

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-3>Q 12

The slope of the line touching both the parabolas $y^{2} = 4 x$ and $x^{2} = - 32 y$ is

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-3>Q 13

Let

O

be the vertex and

Q

be any point on the parabola

x^{2} = 8 y

. If the point

P

divides the line segment

O Q

internally in the ratio

1 : 3

, then the locus of

P

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-3>Q 14

The radius of a circle having minimum area, which touches the curve

y = 4 - x^{2}

and the lines

y = | x |,

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-4>Q 1

Consider the locus of the complex number

z

in the Argand plane is given by

Re (z) - 2 = | z - 7 + 2 i | .

Let

P (z_{1})

and

Q (z_{2})

be two complex number satisfying the given locus and also satisfying

\arg (\frac{z_{1} - (2 + α i)}{z_{2} - (2 + α i)}) = \frac{π}{2}, (α \in R)

then find the minimum value of

P Q

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-4>Q 3

r_{1}, r_{2}

be the length of the perpendicular chords of the parabola

y^{2} = 4 a x

drawn through the vertex, then show that

{(r_{1} r_{2})}^{\frac{4}{3}} = 16 a^{2} (r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}})

Locus of the centre of the circle passing through the vertex and the mid-points of perpendicular chords from the vertex of the parabola y2=4ax is.

Important Questions on Parabola

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Locus of the centre of the circle passing through the vertex and the mid-points of perpendicular chords from the vertex of the parabola $y^{2} = 4 a x$ is.