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1.

The foci of the hyperbola `4x^(2)-9y^(2)-1=0` areA. `(+-sqrt(13),0)`B. `(+-sqrt(13)/6,0)`C. `(0,+-sqrt(13)/6)`D. None of these

Answer» Correct Answer - B
`4x^(2)-9y^(2)=1`
`x^(2)/(1/2)^(2)-y^(2)/(1/3)^(2)=1`
`"eccentricity, e"=sqrt(1+(1/3)^(2)/(1/2)^(2))=sqrt(13)/3`
`"foci "=(+-1/2xxsqrt(13)/3,0)=(+-sqrt(13)/6,0)`
Discussion
2.

What is the locus of points, the difference of whose distances from two points being constant ?A. Pair of straight linesB. An ellipseC. A hyperbolaD. A parabola

Answer» Correct Answer - C
We know that the locus of the difference of whose distances form two points being constant, is a hyperbola.
Discussion
3.

What is the eccentricity of the conic `4x^(2)+9y^(2)=144`?A. `sqrt5/3`B. `sqrt5/4`C. `3/sqrt5`D. `2/3`

Answer» Correct Answer - A
Given equation can be written as
`x^(2)/(36)+y^(2)/(16)=1` ltbtgt This is an ellipse.
`rArra^(2)=36, b^(2)=16`
`e=sqrt(1-b^(2)/a^(2))=sqrt(1-16/36)=sqrt(20/36)=(2sqrt5)/6=sqrt5/3`
Discussion
4.

The eccentricity e of an ellipse satisfies the condition :A. `elt0`B. `0ltelt1`C. `e=1`D. `egt1`

Answer» Correct Answer - B
The eccentricity e of an ellipse satisfies the condition : `0ltelt1`.
Discussion
5.

The sum of focal distances of a point on the ellipse `x^(2)/4+y^(2)/9=1` is:A. 4 unitsB. 6 unitsC. 8 unitsD. 10 units

Answer» Correct Answer - A
Given equation of ellipse is
`x^(2)/36+y^(2)/16=1`
This is an ellipse.
`rArr" "x^(2)/((2)^(2))+y^(2)/((3)^(2))=1`
`rArr" "a=2" and "b=3`
Length of major axis = 2a = 4
Since, we have
Sum of the focal distances of a point on ellipse = length of major axis.
`:." Required Ans = 4 units.
Discussion
6.

Consider the parabola `y=x^(2)+7x+2` and the straight line `y=3x-3`. The two parabolas `y^(2)=4ax" and "x^(2)=4ay` intersectA. at two points on the line y = xB. only at the originC. at three points one of which lies on y + x = 0D. only at (4a, 4a)

Answer» Correct Answer - A
The parabolas `y^(2)=4ax" and "x^(2)=4ay`
They intersect at(0, 0) and (4a, 4a)
These points lie on `y=x`
Discussion
7.

Consider the parabola `y=x^(2)+7x+2` and the straight line `y=3x-3`. The sum of the focal distances of a point on an ellipse is constant and equal to theA. length of equal to theB. length of major axisC. length of major axisD. length of latus rectum

Answer» Correct Answer - B
Discussion
8.

Consider the parabola `y=x^(2)+7x+2` and the straight line `y=3x-3`. The equation of the ellipse whose centre is at origin, major axis is along x-axis with eccentricity `3/4` and latus rectum 4 units isA. `x^(2)/1024+(7y^(2))/64=1`B. `(49x^(2))/1024+(7y^(2))/64=1`C. `(7x^(2))/1024+(49y^(2))/64=1`D. `(x^(2))/1024+(49y^(2))/64=1`

Answer» Correct Answer - B
`"Given, "b^(2)=2a, c^(2)=(3/4)^(2)a^(2)=9/16a^(2)`
`"We know, "a^(2)=b^(2)+c^(2)`
So,
`a^(2)=2a+9/16a^(2)rArr16a^(2)=32a+9a^(2)rArr7a^(2)=32arArra=32/7`
`rArr a = 32/7`
`:. b^(2)=64/7`
Equation of ellipse is `x^(2)/a^(2)+y^(2)/b^(2)=1rArrx^(2)/(32/7)^(2)+y^(2)/(64/7)=1rArr(49x^(2))/(1024)+(7y^(2))/64=1`.
Discussion
9.

Consider the parabola `y=x^(2)+7x+2` and the straight line `y=3x-3`. What is equation of the ellipse having foci `(+-2, 0)` the eccentricity `1/4` ?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`

Answer» Correct Answer - A
`"Foci: "(+-2, 0), e= 1/4`
`c=2, e=1/4=c/arArr2/arArra=8`
`"We know, "a^(2)-b^(2)=c^(2)`
`rArrb^(2)=a^(2)-c^(2)=8^(2)-2^(2)=64-4=60`
`"Eqn of ellipse"rArrx^(2)/a^(2)+y^(2)/b^(2)=1`
`rArr" "x^(2)/64+y^(2)/60=1`
Discussion
10.

If the latusrectum of anellipse is equal to half of minor axis, find its eccentricity.A. `1/2`B. `sqrt3`C. `sqrt3//2`D. `1/sqrt2`

Answer» Correct Answer - C
Let the equation of ellipse be `x^(2)/a^(2)+y^(2)/b^(2)=1`
`"Length of minor axis = 2b"`
`" and length of latus rectum "=(2b^(2))/a`
According to the equation,
`(2b^(2))/a=brArr2b=arArr4b^(2)=a^(2)`
Now, eccentricity of ellipse
`e=sqrt(a^(2)-b^(2))/a`
`e=sqrt(4b^(2)-b^(2))/(2b)=(sqrt3b)/(2b)=sqrt3/2`
`rArr" "e=sqrt3/2`
Discussion
11.

If the latusrectum of anellipse is equal to half of minor axis, find its eccentricity.A. `1//4`B. `1//2`C. `sqrt3//5`D. `sqrt3//2`

Answer» Correct Answer - D
Since, Latusrectum of an ellipse `=(2b^(2))/a`
and minor axis =2b
`:." "b=(2b^(2))/arArra=2b`
`"Also, "e=sqrt(1-b^(2)/a^(2))=sqrt(1-b^(2)/(4b^(2)))=sqrt(3/4)=sqrt3/2`
Discussion
12.

If the latus rectum of an ellipse is equal to the half of minor axis, then find its eccentricity.A. `1/2`B. `sqrt3/2`C. `3/4`D. `sqrt(15)/4`

Answer» Correct Answer - B
Length of latus rectum of an ellipse is `(2b^(2))/a` whereb is semi minor axis. As given , `(2b^(2))/a=b`
`rArr" "2b=arArrb/a=1/2`
`"We know that occentricitye "=sqrt(1-b^(2)/a^(2))=sqrt(1-1/4)=sqrt3/2`
Discussion
13.

The line `2y=3x+12` cuts the parabola `4y=3x^(2)`. What is the equation of the hyperbola having rectum and eccentrieity 8 and `3/sqrt5` respectivly ?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`

Answer» Correct Answer - A
Let the equation of hyperbola be `x^(2)/a^(2)-y^(2)/b^(2)=1`
`"Latus rectum = 8 ="(2b^(2))/arArrb^(2)=4a" …..(i)"`
Also, `b^(2)=a^(2)(e^(2)-1)" [From (i)]"`
`rArr" "4a=a^(2)[(3/sqrt5)^(2)-1]`
`rArr" "a=5&b^(2)=20`
`:." Equation is "x^(2)/25-y^(2)/20=1`
Discussion
14.

The line `2y=3x+12` cuts the parabola `4y=3x^(2)`. Where does the line cut the parabola ?A. 7 square unitB. 14 square unitC. 20 square unitD. 21 square unit

Answer» Correct Answer - C
Equation of line `2y=3x+12` and equations of parabola `4y=3x^(2)`
`=int_(0)^(4)((3x+12)/2-(3x)^(2)/4)dx=(3/4x^(2)+6x-x^(3)/4)_(0)^(4)`
`=3xx4+24-16=36-16=20" sq. units"`
`:." Area enclosed by the parabola, the line and the y axis in first quadrant = 20 sq. units"`
Discussion
15.

The line `2y=3x+12` cuts the parabola `4y=3x^(2)`. Where does the line cut the parabola ?A. `At(-2, 3)` onlyB. At (4, 12) onlyC. At both (-2, 3) and (4, 12)D. Neither at (-2, 3) nor (4, 12)

Answer» Correct Answer - C
Equation of line
`2y=3x+12" ..(1)"`
Equation of parabola
`4y=3x^(2)" …(2)"`
From eqs. (i) and (ii), we get
`2(3x+12)=3x^(2)`
`3x^(2)-6x-24=0`
`x^(2)-2x-8=0`
`(x-4)(x+2)=0`
`:." "x=4`
`" and "x = -2`
Now putting the value of x in eqn (ii)
We get y = 12 and y = 3
Thus, the points (-2, 3) and (4, 12)
Discussion
16.

The line `2y=3x+12` cuts the parabola `4y=3x^(2)`. What ishte eccentricity of rectangular hyper bola ?A. `sqrt2`B. `sqrt3`C. `sqrt5`D. `sqrt6`

Answer» Correct Answer - A
Here, `b^(2)=a^(2)(e^(2)-1)`
For rectangular hyperbola : a = b
`rArrb^(2)=b^(2)(e^(2)-1)`
`rArr" "e^(2)-1=1`
`rArr" "e^(2)=2rArre=+-sqrt2`
`"For hyperbola, "egt1`.
Hence, `e=sqrt2`
Discussion
17.

The focal distance of a point on the parabola `y^2=12 xi s4.`Find the abscissa of this point.A. 1B. `-1`C. `2sqrt2`D. `-2`

Answer» Correct Answer - A
Focal distance of a point `(x_(1), y_(1))` on the parabola is `y^(2)=4ax` is equal distance from directrix x+a=0 is
`x_(1)+a`
`"For "y^(2)=12x,` comparing with `y^(2)=4ax`.
`4a=12rArra=3`
`"so, "x_(1)+2=4`
`rArr" "x_(1)=1`
Discussion
18.

Which one of the following points lies outside the ellipse `(x^(2)//a^(2))+(y^(2)//b^(2))`?A. (a, 0)B. (0, b)C. (-a, 0)D. (a, 0)

Answer» Correct Answer - D
`"The equation of ellipse is "x^(2)/a^(2)+y^(2)/b^(2)-1=0`
The point for which `x^(2)/a^(2)+y^(2)/b^(2)-1=0gt0` is outside ellipse.
Since, at (a, 0)1 + 0 - 1 = 0
It lies on the ellipse.
At (0, b), 0 + 1 - 1 = 0
It lies on the ellipse.
At (-a, 0), 1 + 0 - 1 = 0
It lien on the ellipse.
At (a, 0), `1 + 1 - 1 gt 0`
So, the point (a, b) lies outside the ellipse.
Discussion
19.

Consider an ellipse `x^(2)/a^(2)+y^(2)/b^(2)=1` What is the eauation of parabola whose verted is at (0, 0) and focus is at (0, 2) ?A. `y^(2)+8x=0`B. `y^(2)-8x=0`C. `x^(2)+8y=0`D. `x^(2)-8y=0`

Answer» Correct Answer - C
Focus is (0, 2)
a= -2 and parabola is along y-axis downward
x`x^(2)=4ay`
`x^(2)=-8y`
or `x^(2)+8y=0`
Discussion