MATHS BY RUPESH K JHA... GS ROAD CHRISTIAN BASTI GUWAHATI 1
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TBS-Parabola _Ent
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The Next Class
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Lets Revise the Last Class....it’s not a luxury, it’s a necessity
Q1
Find
(i) The vertex, axis, focus, directrix, length of latusrectum of the parabola x2 + 2y – 3x + 5 = 0.
(ii) The equation of the parabola whose focus is (1, 1) and the directrix is x + y + 1 = 0.
(iii) The equation to the parabola whose focus is (1, –1) and vertex is (2, 1).
(iv) The equation of the directrix of the parabola x2 – 4x – 3y + 10 = 0.
Q2
Find the axis, vertex, focus, directrix and equation of latus rectum of the parabola 9y2–16x–12y–57 = 0
Q3
The equation of the parabola whose focus is (− 3, 0) and the directrix is x + 5 = 0 is:
(A) y2 = 4 (x − 4)
(B) y2 = 2 (x + 4)
(C) y2 = 4 (x − 3)
(D) y2 = 4 (x + 4)
Q4
If (2, 0) is the vertex & y − axis is the directrix of a parabola, then its focus is:
(A) (2, 0)
(B) (− 2, 0)
(C) (4, 0)
(D) (− 4, 0)
Q5.
Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ
internally in the ratio 1 : 3, then the locus of P is
(1) x2 = y
(2) y2 = x
(3) y2 = 2x
(4) x2 = 2y
Q6
Length of the latus rectum of the parabola 25 [(x − 2)2 + (y − 3)2] = (3x − 4y + 7)2 is:
(A) 4
(B) 2
(C) 1/5
(D) 2/5
Q7
Find the all possible values of  such that point P(, ) is outside the parabola y = x2 + x + 1 and inside the
circle x2 + y2 = 50.
(A) (–5, )
(B) (–, )
(C) (–1, 5)
(D) (–5, 5)
Q8
Which one of the following equations parametrically represents equation to a parabolic profile?
Q9.
(A) x = 3 cos t; y = 4 sin t
(B) x2 − 2 = − 2 cos t; y = 4 cos2
(C)
(D) x = 1 − sin t ; y = sin
x = tan t;
y = sec t
t
2
t
t
+ cos
2
2
The points on the parabola y2 = 12x whose focal distance is 4, are
(
) (
(A) 2, 3 , 2, − 3
) (B) (1, 2 3 ) , (1, − 2 3 ) (C) (1, 2), (2, 1) (D) ( 2, 2 3 ) , (3, −2 3 )
Q10
A line y = x + 5 intersect the parabola (y – 3)2 = 8(x + 2) at A & B. Find the length of chord AB.
Q11.
The angle made by a double ordinate of length 8a at the vertex of the parabola y2 = 4ax is :
(A) /3
(B) /2
(C) /4
(D) /6
Q12
Find the locus of the mid-points of the chords of the parabola y2 = 4ax which subtend a right angle at the vertex
of the parabola.
Q13.
For what value of , does the line y = 3x +  touch the hyperbola 9x2 – 5y2 = 45?
DISTRACTIONS MAKE LEARNING HARDER
MATHS BY RUPESH K JHA... GS ROAD CHRISTIAN BASTI GUWAHATI 2
Q14.
Prove that the straight line x + my + n = 0 touches the parabola y2 = 4ax if n = am2.
Q15
Find the range of c for which the line y = mx + c touches the parabola y2 = 8 (x + 2
Q16
Find the set of those value(s) of '' for which the point  7 −
Q17
A parabola y = ax2 + bx + c crosses the x − axis at (, 0) (, 0) both to the right of the origin. A circle also passes
through these two points. Find the length of a tangent from the origin to the circle.

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
5
y2
x2

,   lies inside the ellipse
+
=1.
4
25 16

Q18
Find the set of possible value of  for which point P(, 3) lies on the smaller region of the ellipse
9x2 + 16y2 = 144 divided by the line 3x + 4y = 12.
Q19
Find the length of chord x – 2y – 2 = 0 of the ellipse 4x2 + 16y2 = 64.
Q20
The locus of point of trisections of the focal chords of the parabola, y2 = 4x is:
(A) y2 = x − 1
(B) 9y2 = 4.(3x – 4)
(C) y2 = 2 (1 − x) (D) None of these
Q21
The latus rectum of a parabola whose focal chord is PSQ such that SP = 3 and SQ = 2 is given by:
(A) 24/5
(B)) 12/5
(C) 6/5
(D) 23/5
Q22
If y = 2 x − 3 is a tangent to the parabola y2= 4a  x − 1  , then ' a ' is equal to, where a  0 :

(A) 1
Q23
Q24
Q25
(B) − 1
3
14
3
(C)
(D)
− 14
3
Find the equation of parabola, whose axis is parallel to y-axis and which passes through points
(0, 2), (– 1, 0) and (1, 6).
Find the locus of the middle point of the chords of the parabola y 2 = 4ax which passes
through the focus
The length of the common chord of the parabolas y2 = x and x2 = y is
1
(A) 2√2
(B) 1
(C) √2
(D)
√2
Q26.
2
The point (1, 2 ) is one extremity of focal chord of parabola y = 4 x . The length of this focal chord is
1) 2
Q27
2) 4
3) 6
2
The Cartesian equation of the curve whose parametric equations are x = t + 2t + 3 and y = t + 1 is
2
2
1) y = ( x − 1) + 2 ( y − 1) + 3
2) x = ( y − 1) + 2 ( y − 1) + 5
2
3) x = y + 2
Q28.
4) none of these
4) none of these


2
2
2
The length of the latus-rectum of the parabola 169 ( x − 1) + ( y − 3) = ( 5 x − 12 y + 17 ) is
1)
12
13
2)
14
13
3)
28
13
4)
31
13
Q29.
2
The co-ordinates of a point on the parabola y = 8 x whose focal distance is 4 is
Key.
1
1) ( 2, 4 )
2) ( 4, 2 )
3) ( 2, −6 )
4) ( 4, −2 )
DISTRACTIONS MAKE LEARNING HARDER
MATHS BY RUPESH K JHA... GS ROAD CHRISTIAN BASTI GUWAHATI 3
25
and points are of the form  a,b Then a + b
4
(
2
If focal distance of a point on the parabola y = x − 4 is
Q30.
is equal to
1) 8
2) 4
3) 2
)
4) 0
2
Length of side of an equilateral triangle inscribed in a parabola y − 2 x − 2 y − 3 = 0 whose one angular point
is vertex of the parabola is
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Q31.
1) 2 3
2) 4 3 3) − 3
[Hint: Length is Invariant under change of Axis.
3
4)
2
Q32. Length of the focal chord of the parabola ( y + 3) = −8( x − 1) which lies at a distance 2 units from the vertex
of the parabola is
B) 6 2
A) 8
Q33.
2
2
2
If any point P ( x, y ) satisfies the relation ( 5 x − 1) + ( 5 y − 2 ) =  ( 3x − 4 y − 1) , represents parabola, then
1)  = 1
Q34.
D) 5 3
C) 9
2)   1
4)   2
3)   1
a3 x 2 a 2 x
+
− 2a (a is parameter) is
3
2
64
35
(C) xy =
(D) xy =
105
16
The locus of the vertex of the family of parabolas y =
(A) xy =
105
64
(B) xy =
3
4
Answer
Q1
Q2
3
2
11 
 3 15 
, focus  , − 
8 
8 
2
(i)
vertex   , −
(ii)
x2 – 2xy + y2 – 6x – 6y + 3 = 0
y=
(iii)
axis x =
3
7
, directrix y = – , length of latus rectum = 2.
2
8
4x2 – 4xy + y2 + 8x + 46y – 71 =0
2  61 2   485 2 
−613
485
,  − , ,  −
, , x =
,x = −
3  16 3   144 3 
144
144
Q3)D Q4)C Q5)4 Q6)D Q7)D Q8)B Q9)B Q10)8 2
Q13.) = ± 6
Q11)B Q12)y2 – 2ax + 8a2 = 0
Q15).(– , – 4]  [4, )
Q16
 12 16 
 5 , 5 


Q17
Q19
35
Q20
c
a
D
Q18
4
<<
5
4
17
Q21)A Q22)D
Q24) y2 = 2a(x – a)
Q25C Q26)2 Q27)3 Q28)3 Q29)1 Q30)1 Q31)2 Q32)8 Q33)1 Q34)A
Q23)y = x2 + 3x + 2
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(iv)
y = 5/4