Find the equation of the direct common tangents to the circlesx2+y2+22x−4y−100=0 and x2+y2−22x+4y+100=0

strong{font-weight:bold !important} Given circles are x2+y2+22x−4y−100=0, x2+y2−22x+4y+100=0Centre C1=(−11,2)Centre C2=C2=(−g,f)=(11,−2)Radius r1=g2+f2−cRadius r2=121+4−100=121+4+100=25=5=225=15Distance between the centresC1C2=(22)2+42=500r1+r2=15+5=20=400500>400⇒C1C2>r1+r2∴The circles are disjoint 2 Direct common tangents existExternal center of similitude divides C1C2 in the ratio r1 : r2externally =−r1:r2=−15:5=−3:1∴ECS=−3⋅C2+1⋅C1−3+1=−33−11−2,6+2−2=(22,−4)Direct common tangents pass through ECS, let slope of DCT is m Equation of DCT isy−y1=mx−x1⇒y+4=m(x−22)⇒mx−y−22m−4=0But, it is a tangent to circle (1) ⇒d=r1 i.e., Length of ⊥er from C1(−11,2) to DCT=r1⇒|m(−11)−2−22m−4|m2+1=15⇒|−33m−6|=15m2+1⇒3|11m+2|=15m2+1SOBS(11m+2)2=25m2+1⇒121m2+44m+4=25m2+25⇒96m2+44m−21=0⇒96m2+72m−28m−21=0⇒24m(4m+3)−7(4m+3)=0⇒(4m+3)(24m−7)=0 ⇒m−34,m=724If m=−34Equation of DCT is y+4=−34(x−22)⇒4y+16=−3x+66⇒3x+4y−50=0If m=724Equation of DCT is y+4=724(x−22)⇒24y+96=7x−154⇒7x−24y−250=0

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

Find the equation of the direct common tangents to the circles
$\begin{array}{l} x^{2} + y^{2} + 22 x - 4 y - 100 = 0 and \\ x^{2} + y^{2} - 22 x + 4 y + 100 = 0 \end{array}$

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Given circles are $x^{2} + y^{2} + 22 x - 4 y - 100 = 0,$ $x^{2} + y^{2} - 22 x + 4 y + 100 = 0$
$Centre C_{1} = (- 11, 2)$
$Centre C_{2} = C_{2} = (- g, f) = (11, - 2)$
$Radius r_{1} = \sqrt{g^{2} + f^{2} - c}$
$Radius r_{2} = \sqrt{121 + 4 - 100}$
$\begin{array}{l} = \sqrt{121 + 4 + 100} = \sqrt{25} = 5 \\ = \sqrt{225} = 15 \end{array}$

Distance between the centres
$\begin{array}{l} C_{1} C_{2} = \sqrt{(22)^{2} + 4^{2}} = \sqrt{500} \\ r_{1} + r_{2} = 15 + 5 = 20 = \sqrt{400} \\ \sqrt{500} > \sqrt{400} \Rightarrow C_{1} C_{2} > r_{1} + r_{2} \end{array}$
Question Image
$∴$ The circles are disjoint 2 Direct common tangents exist
External center of similitude divides C₁C₂ in the ratio r₁ : r₂
$externally = - r_{1} : r_{2} = - 15 : 5 = - 3 : 1$
$∴ ECS = \frac{- 3 \cdot C_{2} + 1 \cdot C_{1}}{- 3 + 1}$
$= (\frac{- 33 - 11}{- 2}, \frac{6 + 2}{- 2}) = (22, - 4)$
Direct common tangents pass through ECS, let slope of DCT is m Equation of DCT is
$\begin{array}{l} y - y_{1} = m (x - x_{1}) \\ \Rightarrow y + 4 = m (x - 22) \Rightarrow mx - y - 22 m - 4 = 0 \end{array}$
$But, it is a tangent to circle (1) \Rightarrow d = r_{1} i.e.,$
$Length of ⊥^{er} from C_{1} (- 11, 2) to DCT = r_{1}$
$\begin{array}{l} \Rightarrow \frac{| m (- 11) - 2 - 22 m - 4 |}{\sqrt{m^{2} + 1}} = 15 \\ \Rightarrow | - 33 m - 6 | = 15 \sqrt{m^{2} + 1} \\ \Rightarrow 3 | 11 m + 2 | = 15 \sqrt{m^{2} + 1} \end{array}$
SOBS
$\begin{array}{l} (11 m + 2)^{2} = 25 (m^{2} + 1) \\ \Rightarrow 121 m^{2} + 44 m + 4 = 25 m^{2} + 25 \end{array}$
$\begin{array}{l} \Rightarrow 96 m^{2} + 44 m - 21 = 0 \\ \Rightarrow 96 m^{2} + 72 m - 28 m - 21 = 0 \\ \Rightarrow 24 m (4 m + 3) - 7 (4 m + 3) = 0 \\ \Rightarrow (4 m + 3) (24 m - 7) = 0 \Rightarrow m \frac{- 3}{4}, m = \frac{7}{24} \end{array}$
$If m = \frac{- 3}{4}$
$Equation of DCT is y + 4 = \frac{- 3}{4} (x - 22)$
$\Rightarrow 4 y + 16 = - 3 x + 66 \Rightarrow 3 x + 4 y - 50 = 0$
$If m = \frac{7}{24}$
$Equation of DCT is y + 4 = \frac{7}{24} (x - 22)$
$\Rightarrow 24 y + 96 = 7 x - 154 \Rightarrow 7 x - 24 y - 250 = 0$