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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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Important Questions on Parabola

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JEE Main/Advance

IMPORTANT

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

Let

A

and

B

be two distinct points on the parabola

y^{2} = 4 x

. If the axis of the parabola touches a circle of radius

r

having

A B

as its diameter, then the slope of the line joining

A

and

B

can be

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-3>Q 5

Let

L

be normal to the parabola

y^{2} = 4 x

. If

L

passes through the point

(9, 6)

, then

L

is given by

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-3>Q 9

Let the curve

C

be the mirror image of the parabola

y^{2} = 4 x

with respect to the line

x + y + 4 = 0

. If

A

and

B

are the points of intersection of

C

with the line

y = - 5

, then the distance between

A

and

B

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-3>Q 10

If the normals of the parabola

y^{2} = 4 x

drawn at the end points of its latus rectum are tangents to the circle

(x - 3)^{2} + (y + 2)^{2} = r^{2}

, then the value of

r^{2}

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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 |,

The tangent PT and the normal PN to the parabola y2=4ax at a point P on it meet its axis at points T and N respectively. The locus of the centroid of triangle PTN is a parabola whose

Important Questions on Parabola

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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