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The radius of a circle having minimum area, which touches the curve $y = 4 - x^{2}$ and the lines $y = | x |,$ is

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Important Questions on Parabola

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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 2

Prove that in a parabola the angle

θ

that the latus rectum subtends at the vertex of the parabola isindependent of the latus rectum and lies between

\frac{2 π}{3} & \frac{3 π}{4}

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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}})

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

A chord is normal to a parabola

y^{2} = 4 a x

and is inclined at an angle $θ$ to the axis; prove that the area of the triangle formed by it and the tangents at its extremities is

4 a^{2} \sec^{3} θ {cosec}^{3} θ

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-4>Q 5

From an external point

P

, a pair of tangent lines are drawn to the parabola

y^{2} = 4 x

. If

θ_{1}

and

θ_{2}

are the inclinations of these tangents with the axis of

x

such that

θ_{1} + θ_{2} = \frac{π}{4},

then find the locus of

P

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-4>Q 6

T P

and

T Q

are tangents to the parabola and the normal at

P

and

Q

meet at a point

R

on the curve. Prove that the centre of the circle circumscribing the triangle

T P Q

lies on the parabola

2 y^{2} = a (x - a) .

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-4>Q 7

From an external point

P

, tangents are drawn to the parabola,

y^{2} = 4 a x

; find the equation of the locus of

P

when these tangents make angles

θ_{1}

and

θ_{2}

with the axis, such that

\cos θ_{1} \cos θ_{2}

is a constant

(= μ) .

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Alpha Question Bank for Engineering: Mathematics>Chapter 17 - Parabola>Exercise-4>Q 8

A pair of tangents are drawn to a parabola and are equally inclined to a straight line whose inclination to the axis is

α

; prove that the locus of their point of intersection is the straight line

y = (x - a) \tan 2 α .

The radius of a circle having minimum area, which touches the curve y=4-x2 and the lines y=|x|, is

Important Questions on Parabola

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The radius of a circle having minimum area, which touches the curve $y = 4 - x^{2}$ and the lines $y = | x |,$ is