lf two conics whose equations refer to rect- angular axes are $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
and $a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x + 2 f y + c^{'} = 0$ intersect in four concyclic points, then

h^{'} (a - b) = h (a^{'} - b^{'})

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

h (a - b) = h^{'} (a^{'} - b^{'})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

h (a^{'} - b) = h^{'} (a - b^{'})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

h (a - b^{'}) = h^{'} (a^{'} - b)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $h^{'} (a - b) = h (a^{'} - b^{'})$
$S 1 : a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
$S 2 : a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x + 2 f y + c^{'} = 0$
four points intersects on the curve: $S 1 + k S 2 = 0$
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c + k (a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x + 2 f y + c^{'}) = 0$
Since, they are con-cyclic points
Therefore, coefficient of $x^{2} =$ coefficient of $y^{2}$ and coefficient of $x y = 0$
$a + k a^{'} = b + k b^{'}$
$2 h + 2 k h^{'} = 0$
Eliminating k, we get $h^{'} (a - b) = h (a^{'} - b^{'})$

Suggest Corrections

Similar questions

Q. If two curves whose equations are

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

and

a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x + 2 f^{'} y + c^{'} = 0

intersect in four concyclic points, prove that

\frac{a - b}{h} = \frac{a^{'} - b^{'}}{h^{'}}

Q. If two curves whose equations are

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

and

a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x + 2 f^{'} y + c^{'} = 0

intersect in four concyclic points, then -

Q. Show that the straight lines joining the origin to the other two points of intersection of the curves whose equations are

a x^{2} + 2 h x y + b y^{2} + 2 g x = 0

and

a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x = 0

will be at right angles if

g (a^{'} + b^{'}) - g^{'} (a + b) = 0 .

The line joining the origin to the points of intersection of the curves
$a x^{2} + 2 h x y + b y^{2} + 2 g x = 0$ and $a^{^{'}} x^{2} + 2 h^{^{'}} x y + b^{^{'}} y^{2} + 2 g^{^{'}} x = 0$
will be mutually perpendicular, if

Q. In the equation

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0,

a = b = h = 1

and

f \neq g,

then eccentricity of the conic is

Join BYJU'S Learning Program

lf two conics whose equations refer to rect- angular axes are ax2+2hxy+by2+2gx+2fy+c=0and a′x2+2h′xy+b′y2+2g′x+2fy+c′=0 intersect in four concyclic points, then

lf two conics whose equations refer to rect- angular axes are $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
and $a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x + 2 f y + c^{'} = 0$ intersect in four concyclic points, then