Statement 1 : An equation of a common tangent to the parabola y 2 = 16 3 x and the ellipse 2 x 2 + y 2 = 4 is y = 2 x + 2 3 . Statement 2 : If the line y = m x + 4 3 m , m ≠ 0 is a common tangent to the parabola y 2 = 16 3 x and the ellipse 2 x 2 + y 2 = 4 , then m satisfies m 4 + 2 m 2 = 24 .

Statement 1 is true, Statement 2 is true; statement 2 is a correct explanation for statement 1 .

Statement 1 is false, Statement 2 is true.

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1 .

Statement 1 is true, statement 2 is false.

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The ellipse $E_{1} : \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_{2}$ passing through the point $(0, 4)$ circumscribes the rectangle $R$ . The eccentricity of the ellipse $E_{2}$ is

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Important Questions on Ellipse

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 6

Suppose that the foci of the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1

are

(f_{1}, 0) a n d (f_{2}, 0)

where

f_{1} > 0 a n d f_{2} < 0 .

Let

P_{1} a n d P_{2}

be two parabolas with a common vertex at (0, 0) and with foci at

(f_{1}, 0)

and

(2 f_{2}, 0),

respectively. Let

T_{1}

be a tangent to

P_{1}

which passes through

(2 f_{2}, 0)

and

T_{2}

be a tangent to

P_{2}

which passes through

(f_{1}, 0)

. If

m_{1}

is the slope of

T_{1}

and

m_{2}

is the slope of

T_{2},

then the value of

(\frac{1}{m_{1}^{2}} + m_{2}^{2})

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 7

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 the tangent to a suitable conic (List I) at $(\sqrt{3}, \frac{1}{2})$ is found to be $\sqrt{3} x + 2 y = 4,$ then which of the following options is the only CORRECT combination?

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 8

The ellipse

x^{2} + 4 y^{2} = 4

is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point

(4, 0)

. Then, the equation of the ellipse is

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 9

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point

(- 3, 1)

and has eccentricity

\sqrt{\frac{2}{5}}

is -

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 10

Statement $1 :$ An equation of a common tangent to the parabola $y^{2} = 16 \sqrt{3} x$ and the ellipse $2 x^{2} + y^{2} = 4$ is $y = 2 x + 2 \sqrt{3}$ .

Statement $2 :$ If the line $y = m x + \frac{4 \sqrt{3}}{m}, (m \neq 0)$ is a common tangent to the parabola $y^{2} = 16 \sqrt{3} x$ and the ellipse $2 x^{2} + y^{2} = 4,$ then $m$ satisfies $m^{4} + 2 m^{2} = 24$ .

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 11

An ellipse is drawn by taking a diameter of the circle

{(x - 1)}^{2} + y^{2} = 1

as its semi-minor axis and a diameter of the circle

x^{2} + {(y - 2)}^{2} = 4

as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 12

The locus of the foot of perpendicular drawn from the centre of the ellipse

x^{2} + 3 y^{2} = 6

on any tangent to it is

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 13

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1

The ellipse E1:x29+y24=1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E2 passing through the point 0,4 circumscribes the rectangle R. The eccentricity of the ellipse E2 is

Important Questions on Ellipse

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The ellipse $E_{1} : \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_{2}$ passing through the point $(0, 4)$ circumscribes the rectangle $R$ . The eccentricity of the ellipse $E_{2}$ is