Mathematics Previous year question of Ellipse and Hyperbola for NDA

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Previous year question of Ellipse and Hyperbola for NDA

Previous Year Questions Of Ellipse

Q 2773180946

What is the equation of the ellipse having foci `(±2, 0)` and the eccentricity `1/4 ?`
NDA Paper 1 2017

(A)

`x^2/64 +y^2/60 = 1`

(B)

`x^2/60+y^2/64 = 1`

(C)

`x^2/20+y^2/24 =1`

(D)

`x^2/24+y^2/20 = 1`

Solution:

foci `(pm ae , 0)`

`ae= 2 , e=1/4`

`a=8`

`b^2 =64 (1- 1/16)=60`

`x^2/a^2 +y^2/b^2 =1`

`x^2/64 +y^2/60 =1`

Correct Answer is `=>` (A) `x^2/64 +y^2/60 = 1`

Q 2751080824

If the ellipse `9x^2+16y^2 = 144` intercepts the line `3x+4y = 12` then what is the length of the chord so formed ?
NDA Paper 1 2016

(A)

`5` units

(B)

`6` units

(C)

`8` units

(D)

`10` units

Solution:

`x^2/16+y^2/9 = 1` , Line `x/4+y/3 = 1`

Line intersect to given ellipse

`(4 , 0 ) & ( 0 , 3)`

Length of chord `AB = sqrt((4-0)^2+(3-0)^2)`

`AB = 5`

Correct Answer is `=>` (A) `5` units

Q 2261523425

Consider any point `P` on the ellipse `x^2/(25) + y^2/9 = 1` in the first quadrant. Let `r` and `s` represent its distances from `(4, 0)` and `(-4, 0)` respectively, then `(r + s)` is equal to
NDA Paper 1 2015

(A)

`10` units

(B)

`9` units

(C)

`8` units

(D)

`6` units

Solution:

We have, an ellipse `x^2/(25) + y^2/9 = 1`

Clearly, its foci are `( 4, 0)` and `(- 4, 0)`.

`[ ∵ foci = S (ae, 0), S' (ae, 0)]`

`:. PS + PS' = 2a =` Major axis

`=> r + s = 2(5) = 10` units

Correct Answer is `=>` (A) `10` units

Q 1782145937

What is the sum of the major and minor axes of
the ellipse whose eccentricity is `4//5` and length of
latusrectum is `14.4` units?
NDA Paper 1 2014

(A)

32 units

(B)

48 units

(C)

64 units

(D)

None of these

Solution:

We know that, length of major axes of an

ellipse `= 2a`

and length of minor axes of an ellipse `= 2b`

Given that,

eccentricity of an ellipse `= 4 //5 = e` .........(1)

and length of latusrectum of an ellipse `= 14.4` units

`=> ( 2b^2)/a = 14.4`

` => (b^2)/a = 7.2`

` => b^2 = 7.2 a` .......(2)

Since, eccentricity of an ellipse,

` b^2 = a^2 (1-e^ 2)`

` => 7.2a = a^2 [1 - (4/5)^2 ] ` [from Eqs. (i) and (ii)]

` => 7.2a = a^2 (1- (16)/(25))`

` => 7.2 a = a^2 xx 9/(25)`

` => 9a^2 - 7.2 xx 25a = 0`

` => 9a^2 -36 xx 5a = 0`

` => 9a (a-20) = 0`

` => a = 20`

Put the value of a in Eq. (2), we get

` b^2 =7.2 xx 20`

`=> b^2 = 72 xx 2 = 144`

` => b^2 = (12)^2`

`:. b = 12`

Hence, the sum of the major and minor axes

` = 2a + 2b`

`= 2 (a + b) = 2 (20 + 12)`

` = 2 xx 32`

` = 64` units

Correct Answer is `=>` (C) 64 units

Q 1629167911

What is the length of the latus rectum of an ellipse
`25x^2 + 16y^2 = 400?`

NDA Paper 1 2014

(A)

` (25)/2`

(B)

` (24)/4`

(C)

` (16)/5`

(D)

` (32)/5`

Solution:

Equation of ellipse is `25x^2 + 16y^2 = 400`.

`=> x^2/(16) + y^2/(25) = 1`

Here, `a^2 = 16` and `b^2 = 25`

:. Length of latus rectum `= (2a^2)/b = (2 xx 16)/5 = (32)/5`

Correct Answer is `=>` (D) ` (32)/5`

Q 1783123047

Consider an ellipse, `x^2/a^2 + y^2/b^2 = 1`

What is the area of the greatest rectangle that can be
inscribed in the ellipse?
NDA Paper 1 2014

(A)

`ab`

(B)

`2ab`

(C)

`(ab)/2`

(D)

`sqrt(ab)`

Solution:

Given ellipse, `x^2/a^2 + y^2/b^2 = 1` .........(1)

Let` A (a cos theta , b sin theta)` be any point on an ellipse.

(1st quadrant)

Coordinate of `B`

`= {a cos (pi - theta ), b sin ( pi - theta )}`

`= (-a cos theta , b sin theta )` (2nd quadrant)

and coordinate of `C`

`= {a cos (pi + theta), b sin (pi + theta)`

`= (- a cos theta , - b sin theta )` (3 rd quadrant)

and coordinate of `D`

` = {a cos (2pi - theta), b sin (2pi - theta)}`

`= (a cos theta ,- b sin theta)` (4 quadrant)

Now, area of rectangle `ABCD,`

`Delta = CD xx AD`

`= (a cos theta + a cos theta ) xx (b sin theta + b sin theta )`

` = 2 a cos theta xx 2 b sin theta`

` = 2ab sin 2theta`

Here, area of rectangle will be greatest, when `sin 2theta` have

its maximum value i.e., `sin 2 theta = 1`.

`:. Delta = 2ab xx 1 = 2ab`

Hence, area of greatest rectangle is equal to `2ab`, when

`sin 2theta = 1`.

Correct Answer is `=>` (B) `2ab`

Q 1753223144

Consider an ellipse, `x^2/a^2 + y^2/b^2 = 1`

What is the area included between the ellipse and the
greatest rectangle inscribed in the ellipse?
NDA Paper 1 2014

(A)

`ab (pi - 1)`

(B)

`2ab (pi - 1)`

(C)

`ab (pi - 2)`

(D)

None of these

Solution:

Given ellipse, `x^2/a^2 + y^2/b^2 = 1`

We know that,

Area of the ellipse `x^2/a^2 + y^2/b^2 = 1` is, `Delta ' = pi ab`

`:.` Required area =Area of shaded region

=Area of an ellipse

-Area of greatest rectangle

`= pi ab - 2ab`

` = ab (pi -2)`

Correct Answer is `=>` (C) `ab (pi - 2)`

Q 2328078801

The equation of the ellipse whose vertices are at

`(pm 5,0)` and foci at `(pm 4,0)` is
NDA Paper 1 2013

(A)

`x^2/25 + y^2/9 =1`

(B)

`x^2/9 + y^2/25 =1`

(C)

`x^2/16 + y^2/25 =1`

(D)

`x^2/25 + y^2/16 =1`

Solution:

Given that,

Foci of an ellipse `=( pm 4, 0) = (pm ae, 0)`

`=> ae=4` ...................(i)

and vertices of an ellipse `=( pm 5, 0)= (pm a, 0)`

`=> a=5` .....................(ii)

From Eqs. (i) and (ii),

`e= 4/5` ......................(iii)

Now, we have a relation

`b^2 = a^2 (1-e^2) => b^2 =25 (1- 16/25)`

`=> b^2 = 25 * 9/25 =>b^2 = 9`

`:. b= pm 3`

`:.` Required equation of an ellipse,

`x^2/a^2 + y^2/b^2 =1`

`:. x^2/25 +y^2/9 =1`

Correct Answer is `=>` (A) `x^2/25 + y^2/9 =1`

Q 2308078808

The sum of the focal distances of a point on the

ellipse `x^2/4+y^2/9 =1` is
NDA Paper 1 2012

(A)

`4` units

(B)

`6` units

(C)

`8` units

(D)

`10` units

Solution:

Since. the sum of focal distances of a point on the

ellipse `x^2/a^2 + y^2/b^2 =1` is equal to `2b`. when `b >a`.

`:. a^2=4` and `b^2 = 9 => a =2` and `b=3`

`:.` Sum of the focal distances `= 2 xx 3 = 6` units

Correct Answer is `=>` (B) `6` units

Q 2318078809

The eccentricity e of an ellipse satisfies the
condition
NDA Paper 1 2012

(A)

`e < 0`

(B)

`0 < e < 1`

(C)

`e=1`

(D)

`e > 1`

Solution:

The eccentricity of ellipse lies between `0` and `1` .

Correct Answer is `=>` (B) `0 < e < 1`

Q 2348280103

What is the sum of the focal distances of a point

of an ellipse `x^2/a^2 + y^2/b^2 =1?`
NDA Paper 1 2011

(A)

`a`

(B)

`b`

(C)

`2a`

(D)

`2b`

Solution:

The sum of the focal distances of a point of an ellipse

`x^2/a^2 + y^2/b^2 =1` is `2a` (by property)

Correct Answer is `=>` (C) `2a`

Q 2358280104

If the latusrectum of an ellipse is equal to half its
minor axis, then what is its eccentricity?
NDA Paper 1 2010

(A)

`1/2`

(B)

`sqrt (3)`

(C)

`sqrt(3)/2`

(D)

`1/sqrt(2)`

Solution:

Now, the length of latusrectum of an ellipse `= (2b^2)/a` and

its length of minor axis ` =2b`

According to the question,

`(2b^2)/a =1/2 (2b) => (2b^2)/a = b`

` => 2b = a => 4b^2 = a^2` `( :. b ne 0 )`

`=> 4a^2 (1-e^2) = a^2` `[ :. a^2 = b^2 (1-e^2) ]`

`=> e^2 = 3/4`

`:. e= sqrt (3)/2`

Correct Answer is `=>` (B) `sqrt (3)`

Q 2368280105

The ellipse `x^2/169 + y^2/25 =1` has the same eccentricity

as the ellipse `x^2/a^2 + y^2/b^2 =1` what is the ratio of `a` to `b` ?
NDA Paper 1 2010

(A)

`5/13`

(B)

`13/5`

(C)

`7/8`

(D)

`8/7`

Solution:

Given ellipse is `x^2/169 + y^2/25 =1`

`e= sqrt (1- 25/169) =12/13` `( :. e= sqrt (1- b^2/a^2) )`

Also, be the eccentricity of the ellipse `x^2/a^2 + y^2/b^2 =1`.

According to the question,

`e =sqrt (1-b^2/a^2) => 12/13 = sqrt (1-b^2/a^2)`

`=>b^2/a^2 = 1- 144/169 =25/169`

`:. a/b = 13/5`

Correct Answer is `=>` (B) `13/5`

Q 2388280107

If `(4, 0)` and `(- 4, 0)` are the foci of an ellipse and the
semi-minor axis is `3`, then the ellipse passes
through which one of the following points?
NDA Paper 1 2010

(A)

`(2,0)`

(B)

`(0,5)`

(C)

`(0,0)`

(D)

`(5,0)`

Solution:

Since, the foci of an ellipse are `( 4, 0)` and `( -4, 0)`.

`:. 2 ae = 8 => ae =4`

and `b =3` (semi-minor axis)

We know that, `e= sqrt (1-b^2/a^2)`

`=> (4/a)^2 = (1- 9/a^2)`

`=> 16/a^2 = (a^2 - 9)/a^2 => a^2 = 25`

`:. a =5`

Thus, the equation of an ellipse is

`x^2/25 +y^2/9 =1`

Which is satisfied by `(5, 0)`.
Hence, the ellipse passes through `(5, 0)`.

Correct Answer is `=>` (D) `(5,0)`

Q 2308280108

A circle is drawn with the two foci of an ellipse

`x^2/a^2 + y^2/b^2 =1` at the end of the diameter. What is

the equation of the circle?
NDA Paper 1 2010

(A)

`x^2 +y^2 = a^2 + b^2`

(B)

`x^2 + y^2 =a^2 -b^2`

(C)

`x^2 + y^2 = 2 (a^2 + b^2)`

(D)

`x^2 + y^2 = 2 (a^2 -b^2)`

Solution:

Since, the foci of an ellipse `x^2/a^2 +y^2/b^2 =1` are `(ae , 0 )` and

`(-ae , 0)`

`:.` Equation of circle whose end points of diameter are `(ae ,0 )`

and `(-ae , 0)` is

`(x - ae )(x + ae) + (y - 0)(y - 0) = 0`

`=> x^2 - a^2 e^2 + y^2 =0`

`=> x^2 + y^2 -a^2 (1- b^2/a^2) =0`

`[ :. e^2 = (1-b^2/a^2) ]`

`=> x^2 +y^2 -a^2 + b^2 =0`

`:. x^2 + y^2 = a^2 - b^2`

Correct Answer is `=>` (B) `x^2 + y^2 =a^2 -b^2`

Q 2318380200

What are the equation of the directrices of the
ellipse `25x^2 + 16 y^2 = 400`?
NDA Paper 1 2010

(A)

`3x pm 25 =0`

(B)

`3y pm 25 =0`

(C)

`x pm 15 =0`

(D)

`y pm 25 =0`

Solution:

The equation of ellipse can be rewritten as

`x^2/16 +y^2/25 =1`

Here, `a=4` and `b =5` `( b > a )`

`:.` Equation of the directrices are

` y = pm b/e = pm 5/(sqrt (1- 16/25) ) = pm25/3` `[ :. a^2 = b^2 (1-e^2) ]`

`=> 3y pm 25 =0`

Correct Answer is `=>` (B) `3y pm 25 =0`

Q 2348380203

Let `E` be the ellipse `x^2/9 + y^2/4 =1` and `C` be the circle

`x^2 + y^2 =9`. Let `P= (1,2)` and `Q= (2,1)`, then which
one of the following is correct?
NDA Paper 1 2010

(A)

Q lies inside C but outside E

(B)

Q lies outside both C and E

(C)

P lies inside both C and E

(D)

P lies inside C but outside E

Solution:

For a point `P(1, 2)`

`E = 4(1)^2 + 9(2)^2 - 36`

`= 40 -36 > 0`

and `C = 1^2 + 2^2 -9`

`= 5 -9 < 0`

So, the point `P` lies outside of `E` and inside of `C`.
For a point `Q(2,1)`,

`E = 4 (2)^2 + 9(1)^2 -36`

`= 16 + 9 -36 < 0`

and `(2)^2 + (1)^2 -9 = 4 +1 - 9 < 0`

Hence, the point `Q` lies inside of `E` and `C`.

Correct Answer is `=>` (D) P lies inside C but outside E

Q 2378380206

What is the eccentricity of an ellipse, if its
latusrectum is equal to one-half of its minor axis?
NDA Paper 1 2009

(A)

`1/4`

(B)

`1/2`

(C)

`sqrt(3)/4`

(D)

`sqrt(3)/2`

Solution:

Length of the latusrectum of an ellipse `= (2b^2)/a`

and length of the minor axis `= 2b`

`:. 1/2 (2b ) = (2b^2)/a => a =2b`

`( :. b ne 0` according to the question)

Also, `e =sqrt (1-b^2/a^2)`

`= sqrt (1- b^2/(4b^2)) = sqrt (3/4) = (sqrt (3))/2`

Correct Answer is `=>` (D) `sqrt(3)/2`

Q 2308380208

What is the sum of focal radii of any point on an
ellipse equal to?
NDA Paper 1 2009

(A)

Length of latusrectum

(B)

Length of major axis

(C)

Length of minor axis

(D)

Length of semi-latusrectum

Solution:

We know that, the sum of focal radii of any point on an
ellipse is equal to length of major axis. i.e., sum of focal radii

`=(a+ x) +(a-x)`

`= 2a =` Major axis

Correct Answer is `=>` (B) Length of major axis

Q 2328480301

Consider the ellipse `x^2/a^2 +y^2/b^2 =1 (b > a)`. Then,

which one of the following is correct?
NDA Paper 1 2008

(A)

Real foci do not exist

(B)

Foci are `(pm ae , 0)`

(C)

Foci are `(pm be, 0)`

(D)

Foci are `(0, pm be)`

Solution:

Given, equation of ellipse

`x^2/a^2 +y^2/b^2 =1`

Since , `b > a`

`:.` Foci `= (0, pm be)` (by property)

Correct Answer is `=>` (D) Foci are `(0, pm be)`

Q 2378480306

What is the area of the ellipse `4x^2 + 9y^2 =1`?

NDA Paper 1 2008

(A)

` 6 pi`

(B)

`pi/36`

(C)

`pi/6`

(D)

`pi/sqrt(6)`

Solution:

Given, equation of ellipse

`4x^2 + 9y^2 =1`

`=> x^2/(1/2)^2 + y^2/(1/3)^2 =1`

`:. a=1/2` and `b =1/3`

We know that, the area of an ellipse `x^2/a^2 + y^2/b^2 =1`

`= pi ab = pi (1/2)(1/3) = pi/6`

Correct Answer is `=>` (C) `pi/6`

Q 2339278112

If the ellipse `x^2/169 +y^2/25 =1` has the same

eccentricity as the ellipse `x^2/a^2 + y^2/b^2 =1`, then what

is the ratio of `a` to `b`?
NDA Paper 1 2009

(A)

`5/13`

(B)

`13/5`

(C)

`7/8`

(D)

`8/7`

Solution:

Given ellipse is `x^2/169 + y^2/25 =1`

`:. e= sqrt (1-25/169) = 12/13 ` `( :. e=sqrt (1- b^2/a^2) )`

Also, ellipse is `x^2/a^2 + y^2/b^2 =1`

`:. e = sqrt (1-b^2/a^2)` (according to the question)

`:. 12/13 = sqrt (1-b^2/a^2)`

`=> b^2/a^2 =1 -144/169 = 25/169`

`=> a/b = 13/5`

Correct Answer is `=>` (B) `13/5`

Q 2329801711

In how many points do the ellipse `x^2/4 + y^2/8 =1`

and the circle `x^2 + y^2 = 9` intersect?
NDA Paper 1 2007

(A)

One

(B)

Two

(C)

Four

(D)

None of these

Solution:

The given equation of circle and ellipse are respectively

`x^2 +y^2 =9` ..............(i)

and `x^2/4 +y^2/8 =1` ..............(ii)

From Eqs. (i) and (ii),

`x^2/4 + (9-x^2)/8 =1`

`=>2x^2 +9 -x^2 =8`

`:. x^2 = -1 => (x= pm i)`

Thus, it is clear that circle and ellipse never intersect each other.

Correct Answer is `=>` (D) None of these

Previous Year Questions Of Hyperbola

Q 2761780625

What is the equation of the hyperbola having latus rectum and eccentricity 8 and `3/sqrt5` respectively ?
NDA Paper 1 2016

(A)

`x^2/25 - y^2/20 =1`

(B)

`x^2/40 - y^2/20 =1`

(C)

`x^2/40 - y^2/30 =1`

(D)

`x^2/30 - y^2/25 =1`

Solution:

`(2b^2)/a =8`

`b^2=4a`

and `e= 3/(sqrt 5)`

For Hyperbola

`b^2 = a^2 [e^2-1]`

`4a^2 = a^2 [(3/(sqrt 5))^2 -1]`

`a=5 , b= sqrt 20`

So, equation of hyperbola

`(x^2)/25 -(y^2)/20 =1`

Correct Answer is `=>` (A) `x^2/25 - y^2/20 =1`

Q 2711880729

What is the eccentricity of rectangular hyperbola ? .
NDA Paper 1 2016

(A)

`sqrt2`

(B)

`sqrt3`

(C)

`sqrt5`

(D)

`sqrt6`

Solution:

eccentricity of rectangular hyperbola ` = sqrt2`

Correct Answer is `=>` (A) `sqrt2`

Q 1678180006

The hyperbola `x^2/a^2 - y^2/b^2 = 1` passes through the
point `(3 sqrt(5) , 1)` and the length of its latus rectum is
`4/3` units. The length of the conjugate axis is
NDA Paper 1 2015

(A)

2 units

(B)

3 units

(C)

4 units

(D)

5 units

Solution:

Since, hyperbola passes through `(3sqrt(5) , 1)`

`:. (3sqrt(5))^2/a^2 - 1/b^2 = 1`

`=> (45)/a^2 - 1/b^2 = 1`

` => 45b^2 - a^2 = a^2b^ 2` ........(1)

Also, ` (2b^2)/a = 4/3 => 6b^2 = 4a`

` => a = (6b^2)/4` ..........(2)

On putting the value from Eq. (ii) in Eq. (i), we get

` 45b^2 - ((6b^2)/4)^2 = ((6b^2)/4)^2b^2`

` => 45b^2 - (36 b^4)/(16) = (36 b^4 .b^2)/(16)`

` => 45b^2 = (36 b^4)/(16) [b^2 -1]`

` => 45 xx 16 = 36b^2 (b^2 -1) => b^4 +b^2 =20`

`:. b=2`

`:. 2b = 4 = `length of conjugate axis.

Correct Answer is `=>` (C) 4 units

Q 2231123922

The eccentricity of the hyperbola `16 x^2
- 9y^2 = 1` is
NDA Paper 1 2015

(A)

`3/5`

(B)

`5/3`

(C)

`4/5`

(D)

`5/4`

Solution:

Given, `x^2/(1//16) - y^2/(1//9) =1`

`:.` Eccentricity, `e^2 = 1 + b^2/a^2 = 1 + (1//9)/(1//16) = 1 + (16)/9 = (25)/9`

` => e = 5/3 `

Correct Answer is `=>` (B) `5/3`

Q 2378878706

The foci of the hyperbola `4x^2 - 9 y^2 - 1 = 0` are
NDA Paper 1 2013

(A)

`(pm sqrt (13) , 0)`

(B)

`(pm sqrt(13)/6 , 0)`

(C)

`(0, pm sqrt(13)/6)`

(D)

None of these

Solution:

Given equation of hyperbola is

`4x^2 - 9y^2 =1`

`=> x^2/(1/4) - y^2/(1/9) =1`

Here, `a^2 = 1/4` and `b^2 = 1/9`

`:.` Foci of the hyperbola `= (pm ae, 0)`

`= (pm a (sqrt (a^2 + b^2) )/a , 0) = (pm sqrt (a^2 + b^2) ,0 )`

` [ :. b^2 = a^2 (e^2 -1)]`

`=(pm sqrt (1/4 +1//0) , 0 ) = (pm sqrt(13)/6 , 0)`

Correct Answer is `=>` (B) `(pm sqrt(13)/6 , 0)`

Q 2378078806

The difference of focal distances of any point on a
hyperbola is equal to
NDA Paper 1 2013

(A)

latusrectum

(B)

semi-transverse axis

(C)

transverse axis

(D)

semi-latusrectum

Solution:

By definition of hyperbola,

A hyperbola is the set of points in a plane, where distances from
two fixed points in the plane have a constant difference i.e,
'Transverse axis'. The two fixed points are the foci of the
hyperbola.

Correct Answer is `=>` (C) transverse axis

Q 2388480307

If equation of the hyperbola with eccentricity `3 /2`
and foci at `(pm 2, 0)` is `5x^2 - 4y^2 = k^2` , then what is
the value of `k`?
NDA Paper 1 2008

(A)

`4/3`

(B)

`3/4`

(C)

`(4/3)/sqrt(5)`

(D)

`(3/4)sqrt (5)`

Solution:

Given equation of hyperbola

`5x^2 -4y^2 =k^2`

`=> x^2/(k^2/5) - y^2/(k^2/4) =1`

`:. a = k/sqrt(5)` and `b = k/2`

Since, the eccentricity and foci are `3/2` and `(pm 2, 0)` respectively

`:. e =3/2` and `pm ae =2`

`=> k/sqrt(5) * 3/2 =2`

`:. k = 4/3 sqrt(5)`

Correct Answer is `=>` (C) `(4/3)/sqrt(5)`

Q 2339101912

If the eccentricity and length of latusrectum of a

hyperbola are `sqrt(13)/3` and `10/3` units respectively , then

what is the length of the transverse axis ?
NDA Paper 1 2007

(A)

`7/2 ` units

(B)

`12`units

(C)

`15/2` units

(D)

`15/4` units

Solution:

Given that, eccentricity `e = sqrt(13)/3` and length of

latusrectum `= 10/3`

`=> (2b^2)/a =10/3 => b^2 = (5a)/3` ................(i)

We know that, `b^2 = a^2(e^2 - 1)`

`=> (5a)/3=a^2 (13/9 -1)`

`=> (5a)/3 = (4a^2)/9`

`=> 4a^2 -15 a =0`

`=> a(4a -15) =0`

`=> a =15/4` `( :. a ne 0 )`

`:.` Length of transverse axis `= 2a = (2 xx 15 )/4 = 15/2` units

Correct Answer is `=>` (C) `15/2` units