If two tangents to the parabola y2=4ax from a point P make an angle θ1 and θ2 with the axis of the parabola, then find the locus of P if cosθ1cosθ2=λ (a constant)

BACK

Grade upto college level Analytical Geometry

If two tangents to the parabola y²=4ax from a point P make an angle θ₁ and θ₂ with the axis of the parabola, then find the locus of P if cosθ₁cosθ₂=λ (a constant)

Pranjal K

12 Years agoGrade upto college level

1 Answer

Pratik Tibrewal

Approved Tutor Answer12 Years ago

Let the equation of tangent in slope form be y= mx + a/m

Let point P be (h,k)

hence this point will satisfy the line

there fore: k = mh + a/m or mh^2 - km + a = 0 (a quadratic in 'm' which has two roots m1 (tan(a)and m2 tan(b))

gicen cos(a) . cos(b) = K;

from quadratic product of roots: tan(a) . tan(b) = a/h^2 -------- (i)

sum of roots : tan(a) + tan(b) = k/h^2

now sec^2(a). sec^2(b) =(1+tan^2(a))(1+tan^2(b)) = 1 + tan^2(a) + tan^2(b) + tan^2(a) . tan^2(b) =

= 1 + (tan(a) + tan(b))^2 - 2tan(a) . tan(b) + tan^2(a). tan^2(b)

= 1 + k^2/h^4 - 2a/h^2 + a^2/h^4

now cos (a) . cos(b) = 1/sec(a). sec(b) = K

Thanks and Regards,

Pratik Tibrewal

askiitian faculty,

BTech, IITG

The locus of the point represented by x= t^2 +t+1 and y=t^-t+1 is

analytical geometry

1 Answer Available

Last Activity: 4 Years ago

show that the centroids of the triangles of which three perpendiculars lie along the lines y=m1x,y=m2x,y=m3x lie on y(3+m2m3+m3m2+m1m2)=x(m1+m2+m3+3m1m2m3)

analytical geometry

1 Answer Available

Last Activity: 4 Years ago

show that the equation (ab-h²)(ax²+2hxy+by²+2gx+2fy)+af²+bg²-2fgh=0 represent a pair of straight lines and that these straight lines form a rhombus with the lines ax²+2hxy+by²=0

analytical geometry

1 Answer Available

Last Activity: 4 Years ago

A circle has its center on straight line 5x-2y+1=0 and cuts the x axis at two points whose absciasse are -5,3 find equation of the circle and it's radius.

analytical geometry

1 Answer Available

Last Activity: 4 Years ago

Prove that ax^2+by^2+cz^2 +2ux+2vy+2wz+d=0 represent a cone if u^2/a+v^/b+w^2/c=d

analytical geometry

1 Answer Available

Last Activity: 4 Years ago

View all Questions

If two tangents to the parabola y2=4ax from a point P make an angle θ1 and θ2 with the axis of the parabola, then find the locus of P if cosθ1cosθ2=λ (a constant)

Pranjal K

1 Answer

Pratik Tibrewal

Other Related Questions on analytical geometry

The locus of the point represented by x= t^2 +t+1 and y=t^-t+1 is

show that the centroids of the triangles of which three perpendiculars lie along the lines y=m1x,y=m2x,y=m3x lie on y(3+m2m3+m3m2+m1m2)=x(m1+m2+m3+3m1m2m3)

show that the equation (ab-h²)(ax²+2hxy+by²+2gx+2fy)+af²+bg²-2fgh=0 represent a pair of straight lines and that these straight lines form a rhombus with the lines ax²+2hxy+by²=0

A circle has its center on straight line 5x-2y+1=0 and cuts the x axis at two points whose absciasse are -5,3 find equation of the circle and it's radius.

Prove that ax^2+by^2+cz^2 +2ux+2vy+2wz+d=0 represent a cone if u^2/a+v^/b+w^2/c=d

If two tangents to the parabola y²=4ax from a point P make an angle θ₁ and θ₂ with the axis of the parabola, then find the locus of P if cosθ₁cosθ₂=λ (a constant)