Question

C1:x2+ y2 = 25, C2 : x2 + y2 - 2x - 4y - 7 = 0 be two circles intersecting at the
points A and B. Tangents at A and B to the circle C1 intersect at which point?​

Answer 1

Answer:

Intersection point  $(x_{1} ,y_{1} ) = (\frac{25}{9} , \frac{50}{9} )$

Step-by-step explanation:

Given Data:

$C1 : x^{2} + y^{2} - 25 = 0\\C2 : x^{2} + y^{2} - 2x - 4y -7 = 0$

To Find:

Tangents at A and B to the circle C1 intersect at which point?

Solution:

• Equation of common chord AB is

 C1 − C2 = 0

$(x^{2} + y^{2} - 25) - (x^{2} + y^{2} - 2x- 4y-7) = 0$

$x^{2} + y^{2} - 25- x^{2} - y^{2} + 2x + 4y + 7 =0\\\\2x+4y-18=0\\\\x+2y-9=0...(i)$

• Now tangents are made at A and B.

• If tangents meet at P($x_{1},y_{1}$), then AB is chord of contact with respect to P equation of chord of contact is

T = 0

$xx_{1} +yy_{1} -25=0...(ii)$

Compare with (i) and (ii)

$\frac{x_{1} }{1} = \frac{y_{1} }{2} = \frac{- 25}{- 9}$

$x_{1} =\frac{25}{9} \\\\y_{1} = \frac{50}{9}$

$(x_{1} ,y_{1} ) = (\frac{25}{9} , \frac{50}{9} )$

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Answer 2

The tangents at A and B to circle C1 intersect at the point (25/9, 50/9).

Given:

Equation of circle C1 = $x^{2}+y^{2} = 25$

Equation of circle C2 = $x^{2}+y^{2} - 2x - 4y - 7 = 0$

C1 and C2 intersect at points A and B

To find:

The point of intersection of tangents at A and B to the circle C1.

Solution:

The equation of the common chord AB is given by

C1 - C2 = 0

=> $(x^{2}+y^{2} - 25) - (x^{2}+y^{2} - 2x - 4y - 7) = 0$

=> $x^{2}+y^{2} - 25 - x^{2}-y^{2} + 2x +4y +7 = 0$

=> $x + 2y - 9 = 0$ ...(i)

According to the given question, tangents are made at points A and B.

Let us assume that the tangents meet at a point P $(x_{1} ,y_{1})$.

=> AB is the chord of contact concerning P.

The equation for the chord of contact is given by

$xx_{1} + yy_{1} - 25 = 0$...(ii)

By comparing (i) and (ii), we get the equation

$\frac{x_{1} }{1} + \frac{y_{1} }{2} = \frac{-25}{-9}$

=> $x_{1}$ = 25/9

    $y_{1}$ = 50/9

Hence, the tangents at A and B to circle C1 intersect at the point (25/9, 50/9).

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