Let O be the origin, OP and OQ be two perpendicular chords of equal length of the circle x2+y2-4 x+8 y=0. Let m1 and m2 be the slopes of chords OP and OQ. Evaluate |(m1/m2)|, where m1m2.

Question

Let O be the origin, OP and OQ be two perpendicular chords of equal length of the circle x2+y2-4 x+8 y=0. Let m1 and m2 be the slopes of chords OP and OQ. Evaluate |(m1/m2)|, where m1>m2. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/a210e0f4019fed4c5bec9da83bc8fa44-.png" /> x2+y2-4 x+8 y=0 Leftrightarrow(x-2)2+(y+4)2=20 Centre C =(2,-4) Slope of OC =(-4-0/2-0)=-2 Let OP have slope m . Then tan 45°=|(m+2/1-2 m)| Leftrightarrow m+2=±(1-2 m) Leftrightarrow 3 m=-1 or m=3 ⇒ m1=3, m2=-(1/3) Leftrightarrow|(m1/m2)|=|(3/(-(1)3))|=9

Q. Let O be the origin, OP and OQ be two perpendicular chords of equal length of the circle x2+y2−4x+8y=0. Let m1​ and m2​ be the slopes of chords OP and OQ. Evaluate ∣∣​m2​m1​​∣∣​, where m1​>m2​.

x2+y2−4x+8y=0 ⇔(x−2)2+(y+4)2=20 Centre C=(2,−4) Slope of OC=2−0−4−0​=−2 Let OP have slope m. Then tan45∘=∣∣​1−2mm+2​∣∣​ ⇔m+2=±(1−2m) ⇔3m=−1 or m=3 ⇒m1​=3,m2​=−31​ ⇔∣∣​m2​m1​​∣∣​=∣∣​(−31​)3​∣∣​=9

Q. Let $O$ be the origin, $OP$ and $OQ$ be two perpendicular chords of equal length of the circle $x^{2} + y^{2} - 4 x + 8 y = 0$ . Let $m_{1}$ and $m_{2}$ be the slopes of chords $OP$ and $OQ$ .
Evaluate $∣ ∣ \frac{m _{1}}{m _{2}} ∣ ∣$ , where $m_{1} > m_{2}$ .