Chord A B of the circle x2+y2=100 passes through the point (7,1) and subtends an angle of 60° at the circumference of the circle. If m1 and m2 are the slopes of two such chords then the value of m1 m2 is

Question

Chord A B of the circle x2+y2=100 passes through the point (7,1) and subtends an angle of 60° at the circumference of the circle. If m1 and m2 are the slopes of two such chords then the value of m1 m2 is (A) -1 (B) 1 (C) 7 / 12 (D) -3

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/055417456d630a835b7088183d0fa3f1-.png" /> Let the slope of the chord through point (7,1) be m. Thus, equation of line is y-1=m(x-7) or m x-y+1-7 m=0 Perpendicular distance from (0,0)=(r/2) ⇒ (|7 m-1|/√1+m2)=5 ⇒ (7 m-1)2=25(1+m2) ⇒ 49 m2-14 m+1=25+25 m2 ⇒ 24 m2-14 m-24=0 ⇒ m1 m2=-1

Q. Chord $A B$ of the circle $x^{2} + y^{2} = 100$ passes through the point $(7, 1)$ and subtends an angle of $6 0^{\circ}$ at the circumference of the circle. If $m_{1}$ and $m_{2}$ are the slopes of two such chords then the value of $m_{1} m_{2}$ is

$- 1$

$1$

$7/12$

$- 3$

Q. Chord AB of the circle x2+y2=100 passes through the point (7,1) and subtends an angle of 60∘ at the circumference of the circle. If m1​ and m2​ are the slopes of two such chords then the value of m1​m2​ is

−1

1

7/12

−3

Let the slope of the chord through point (7,1) be m. Thus, equation of line is y−1=m(x−7) or mx−y+1−7m=0 Perpendicular distance from (0,0)=2r​ ⇒1+m2​∣7m−1∣​=5 ⇒(7m−1)2=25(1+m2) ⇒49m2−14m+1=25+25m2 ⇒24m2−14m−24=0 ⇒m1​m2​=−1

Q. Chord $A B$ of the circle $x^{2} + y^{2} = 100$ passes through the point $(7, 1)$ and subtends an angle of $6 0^{\circ}$ at the circumference of the circle. If $m_{1}$ and $m_{2}$ are the slopes of two such chords then the value of $m_{1} m_{2}$ is

$- 1$

$1$

$7/12$

$- 3$