1. The equations of the common tangents to the parabola \[y=x^{2}\] and
\[y=-\left(x-2\right)^{2}\] are
a) y = 4(x – 1)
b) y = 0
c) y = –4(x – 1)
d) Both a and b
Explanation: Coordinates of any point on y = x2 is of the form (t, t2)
2. The tangent PT and the normal PN to the parabola \[y^{2}=4ax\] at a point P on it meet its axis at T and N respectively. The locus of the centroid of the triangle PTN is a parabola whose
a) vertex is (2a/3,0)
b) directrix is x=0
c) focus is (a, 0)
d) Both a and c
Explanation: Equation of the tangents and normal at a point P(at2 , 2at) to the parabola y2 = 4ax are ty = x + at2 and y = –tx + 2at + at3
3. y = mx bisects two distinct chords drawn
from (4, 4) on \[y^{2}=4x\] if
a) m = 1
b) m = 0
c) m = 1/2
d) All of the Above
Explanation: Any point on y = mx is (t, mt)
Equation of the chord of the parabola y2 = 4x having (t, mt) as the mid point is
4. All chords of the curve \[3x^{2}-y^{2}-2x+4y=0\] which subtend a right angle at the origin pass through
a) (1, –2)
b) the point of intersection of the lines y + 2x = 0 and
x = 1
c) the vertex of the parabola \[x^{2}-2x-4y-7=0\]
d) All of the Above
Explanation: Let y = mx + c be a chord of the given curve. Equation of the pair of lines through the origin and the points of intersection of the chords and the curve is
5. Let P be the point on the parabola y2 =
4x which is at a shortest distance from the centre S of the
circle \[x^{2}+y^{2}-4x-16y+64=0\]
Let Q be the point on the circle dividing the line segment
SP internally, then
a) \[SP=2\sqrt{5}\]
b) the slope of the tangent to the circle at Q
is\[\frac{1}{2}\]
c) the x-intercept of the normal to the parabola at P is 6.
d) All of the Above
Explanation: Coordinates of S are (2,8)
6. The axis of a parabola is along the line y=x and the distance of its vertex from
origin is \[\sqrt{2}\] and that from its focus is \[2\sqrt{2}\] if vertex and focus both lie in the first quadrant
,then the equation if the parabola
a) \[\left(x+y\right)^{2}=\left(x-y-2\right)\]
b) \[\left(x-y\right)^{2}=\left(x+y-2\right)\]
c) \[\left(x-y\right)^{2}=4\left(x+y-2\right)\]
d) \[\left(x-y\right)^{2}=8\left(x+y-2\right)\]
Explanation:
7. The normal \[y=mx-2am-am^{3}\] to the parabolas \[y^{2}=4ax\] subtends a right angle at the vertex if
a) m=1
b) \[m=\sqrt{2}+1\]
c) \[m=\pm\sqrt{2}\]
d) \[m=\sqrt{2}-1\]
Explanation:
8. An equation of a tangent common to the parabolas \[y^{2}=4x\] and \[x^{2}=4y\]
a) x – y + 1 = 0
b) x + y – 1 = 0
c) x + y + 1 = 0
d) y = 0
Explanation:
9. The coordinates of the end point of the latus rectum
of the parabola \[\left(y-1\right)^{2}=2\left(x+2\right)\] , which does not
lie on the line 2x + y + 3 = 0 are
a) (–2, 1)
b) (–3/2, 1)
c) (–3/2, 2)
d) (–3/2, 0)
Explanation:
10. The point of contact of the tangent to the parabola y2 =9x which passes through
the point (4,10) and makes an angle \[\theta \] with the axis of the parabola such that \[\tan\theta >2\] is
a) (4/9, 2)
b) (36, 18)
c) (4, 6)
d) (1/4, 3/2)
Explanation:
