The equations of two sides of a variable triangle are x = 0 and y = 3 , and its third side is a tangent to the parabola y 2 = 6 x . The locus of its circumcentre is [2023]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
The equations of two sides of a variable triangle are $x = 0$ and $y = 3,$ and its third side is a tangent to the parabola $y^{2} = 6 x$ . The locus of its circumcentre is [2023]

1 $4 y^{2} - 18 y + 3 x + 18 = 0$

2 $4 y^{2} - 18 y - 3 x + 18 = 0$

3 $4 y^{2} + 18 y + 3 x + 18 = 0$

4 $4 y^{2} - 18 y - 3 x - 18 = 0$

Ans.
(1)

$Equation of the parabola is y^{2} = 6 x .$

Here $4 a = 6$

$a = \frac{3}{2}$

Equation of tangent is $y = m x + \frac{a}{m}$

$\Rightarrow y = m x + \frac{3}{2 m}$ ...(i)

Putting $x = 0$ in (i), we get $A \equiv (0, \frac{3}{2 m})$

Putting $y = 3$ in (i), we get $B \equiv (\frac{6 m - 3}{2 m^{2}}, 3)$

The centre of the circle will lie on the line $A B$ as midpoint.

$∴$ $h = \frac{6 m - 3}{4 m^{2}}, k = \frac{3 + 6 m}{4 m}$ $\Rightarrow m = \frac{3}{4 k - 6}$

On substituting $h = \frac{6 m - 3}{4 m^{2}}$ to eliminate $m$ , we get $3 h = 2 (- 2 k^{2} + 9 k - 9)$

$4 k^{2} - 18 k + 3 h + 18 = 0$

So, locus is $4 y^{2} - 18 y + 3 x + 18 = 0$

Want Us to Email you About Special Offers & Updates?