# Parabola - 6

__E-3__ IF Three distinct and real normals can be drawn to y^{2}=8x from the point (a,0) then show that a>4.

ans:- Equation of normal in terms of m is y=mx-4m-2m^{3}=0

it passes through (a,0) then am-4m-2m^{3}=0

m(a-4-2m2)=0

= m=0, m2=

for three distinct normal , (a-4)>0

=a>4

E-4 if y+b=m,(x+a) and y+b=m(x+a) are two tangents to the parabola y^{2}4ax, then show that m1m2=-1

which lies on the directrix x+a=0. hence th etwo tangents intersect on directrix which we know is the locus of perpendicular tangents .

hence m1.m2=-1

E-5 show that the parametric representation (2+t_{2},2t+1)

represents a parabola with vertex at (2,1).

Ans:- x=2+t^{2}, y=2t+1.

Eliminating t= we get (y-1) ^{2} =4(x-2)

i.e, a parabola with vertex at (2,1).

Ans: Equation of the given parabola can be written as,

9x^{2}+12x+4+18y-18=0

i.e, (3x+2)^{2}=18(y-1)

Equation of tangent to the above parabola can be written as-

IF the tangent passes through (0,1) then we have,

0.m^{2} -4m-3=0

gives,m=-

hence equation of the required lines are,

i.e, 12x+9y-1=0

and y-1=0

E-8: the tangents to the parabola y^{2} =4ax at p(at^{2},2at) and q(at_{2}^{2},2at^{2}) intersect at r. prove that the area of the triangle

PQR is

Ans:

equation of tangents at p(at^{2},2at1) and Q(at_{2}^{2},2at_{2})

are, t_{1}y=x+at1^{2}…………(1)

and t_{2}y=x+at_{2}^{2}

since point of intersection of (1) and (2) is r(at_{1}t_{2}) (at_{1}t_{2},a(t_{1}+t_{2})

E-6 the normals with slopes m1, m2, and m3 are drawn from apoint p not on the axis of the parabola y^{2}

=4ax ifv m1 and m2= results in the locus of p being a part of parabola, find the value of

Ans:

Any normal of the parabola y^{2}=4x with slope m is

y=mz-2m-m^{3}

thus, m1.m2.m3=-k

m_{3}=-k(m1m2=

=

m_{3} is aroot of(1) then,

=k3+(2-h)k^{2}-k^{3}=0

locus of p(h,k) is,

y^{3}+(2-x)y^{2}-y^{3}=0

(p does not lie on the axis of the parabola)

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

it is a part of the parabola y^{2}=4x

then ^{2}=4

and -2^{2}+^{3}=0

=-2=0

=2

E-7: find the equation of aline which touches the parabola

9x^{2}+12x+18y-14=0

and passes through the point (0,1).

__AREA OF TRAINGLE__: .

Expanding with respect to first row-

.

E-9: Find the length of the normal chord to the parabola y^{2}=4x which substends a right angle

at the vertex Ans:- a=1 for parabola PQ being normal chord.

PQ substend a right angle at vertex,

E - 10. An equilateral triangle SAB is inscribed in the parabola y2 = 4ax Having its focus at ’S’.

It chord AB lies towards the left of S. Then find the side length of this triangle.

__Ans__:

let A=(at_{1}^{2},2at_{1}),B=(at_{2}^{2}-2at_{1})

we have

Q��� Prove that 9x2 - 24xy + 16y2 - 20x - 15 y -

60 = 0 represents a parabola. Also find its focus and directrix.

__Ans__: Here h^{2}-ab=(-12)^{2}-9x16=144-144=0 also

the equation represents a parabola Now, the equation is(3x-4y)^{2}=5(4x+3y+12) clearly

, the lines 3x-4y=0 and 4x+3y+12=0 are perpendicular to each other.so, let

the equation of the parabola becomes -

Now if

then we havefrom the equations of tranformation in(1)

The equation of directrix is,

The directrix is