wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If tangents are drawn at the points A(1,2),B(4,4) and a variable point C lies on the parabola y2=4x, then the locus of the orthocentre of triangle formed by these tangents is

A
x=0
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
x1=0
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
x+1=0
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
y=0
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is C x+1=0
For the given parabola a=1
Now,
A(1,2)(t21,2t1)t1=1B(4,4)(t22,2t2)t2=2
Let C(t2,2t)

Intersection point of tangents at A and B
=(t1t2,(t1+t2))(2,1)
Let P(2,1)

Intersection point of tangents at C and B
=(2t,t2)
Let R(2t,t2)

Intersection point of tangents at C and A
=(t,t+1)
Let Q(t,t+1)


Slope of QR=1t
Slope of PQ=1

Hence, the equation of altitiude through R on PQ is
y+2t=(x+2t)x+y+t+2=0(1)
The equation of altitiude through P on QR is
y+1=t(x+2)(2)

Therefore, the orthocentre is the point of intersection of equation (1) and (2)
xt2=tx2t1x(t1)+t1=0(x+1)(t1)=0x+1=0 (t1)

The locus of orthocentre is
x+1=0


Alternate Solution:
For any triangle formed by the tangents drawn at t1,t2,t3 on the parabola y2=4ax it's orthocentre will always lies on directrix, i.e. x+a=0
Hence, the locus is x+1=0

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
Join BYJU'S Learning Program
CrossIcon