Question

If the line y = mx + c and r* + y = a intersects at A and B and AB = 2k. then show
that c = (1 + m2)(a2 - k2)​

Answer 1

For real solution

The intersection points are

x

2

+(mx+c)

2

=a

2

and (

m

y−c

)

2

+y

2

=a

2

Solving each

x

2

(1+m

2

)+(2mc)x+(c

2

−a

2

)=0

y

2

(1+m

2

)−2cy+(c

2

−m

2

a

2

)=0

For real solution, discriminant ≥0

D

1

=(2mc)

2

−4(1+m

2

)(c

2

−a

2

)

D

2

=(2c)

2

−4(1+m

2

)(c

2

−m

2

a

2

)

D

1

≥0: 4m

2

c

2

−4[c

2

−a

2

+m

2

c

2

−m

2

a

2

]≥0

4a

2

(1+m

2

)−4c

2

≥0

a

2

(1+m

2

)≥c

2

a

2

(1+m

2

)

≥∣c∣

D

2

≥0: 4c

2

−4[c

2

−m

2

a

2

−m

2

c

2

−m

4

a

2

]≥0

−4[(m

2

+1)(−m

2

a

2

)+m

2

c

2

]≥0

−(m

2

+1)(m

2

a

2

)+m

2

c

2

≤0

c

2

≤(m

2

+1)a

2

∣c∣≤

(m

2

+1)a

2