Co- ordinates of point P and Q are (-3,2) and (5,3) then the product of ordinates of P and ordinate Q will be____________
Answers
Answer:
Let (at
2
,2at
1
),(at
2
2
,2at
2
),(at
3
2
,2at
3
) be coordinates of the point P, Q, R respecting which lie on the curve y
2
=4ax.
Now, equation of the normal to the curve at P is,
y−2at
1
=
2a
−2at
1
(x−at
1
2
) ……….(i)
R is a point which lies at the normal at P.
∴ Substituting y=2at
3
, x=at
3
2
in equation (1)
2at
3
−2at
1
=−t
1
(at
3
2
−at
1
2
)
2a(t
3
−t
1
)=−at
1
(t
3
2
−t
1
2
)
2(t
3
−t
1
)=−t
1
(t
3
−t
1
)(t
3
+t
1
)
2=−t
1
(t
3
+t
1
)
⇒2=−t
1
t
3
−t
1
2
……….(ii)
Now, the equation of the normal to the curve at Q is
y−2at
2
=−
2a
2at
2
(x−at
2
2
) ………..(iii)
R is a point which lies at the normal at Q.
Substituting y=2at
3
,x=at
3
2
in equation (iii)
2at
3
−2at
2
=−t
2
(at
3
2
−at
2
2
)
2a(t
3
−t
2
)=−at
2
(t
3
−t
2
)(t
3
+t
2
)
2=−t
2
(t
3
+t
2
)
⇒2=−t
2
t
3
−t
2
2
……..(iv)
Multiplying equation (ii) with t
2
,
2t
2
=−t
1
t
2
t
3
−t
1
2
t
2
………..(v)
Multiplying equation (iv) with t
1
,
2t
1
=−t
1
t
2
t
3
−t
2
2
t
1
…….(vi)
Subtracting (v) and (vi), we get
2t
2
−2t
1
=−t
1
2
t
2
+t
2
2
t
1
2(t
2
−t
1
)=−t
1
t
2
(t
1
−t
2
)
2(t
2
−t
1
)=t
1
t
2
(t
2
−t
1
)
⇒t
1
t
2
=2
Now, the product of ordinates of P and Q=2at
1
⋅2at
2
=4a
2
t
1
t
2
=4a
2
(2)
=8a
2
.