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Byju's Answer

Standard XII

Mathematics

Parametric Form of Tangent: Ellipse

Circles C1 an...

Question

Circles $C_{1}$ and $C_{2}$ , of radii r and R respectively, touch each other as shown in figure. The line l, which is parallel to the line joining the centres of $C_{1}$ and $C_{2}$ , is tangent to $C_{1}$ at P and intersects $C_{2}$ at A,B. If $R^{2} = 2 r^{2}$ , then $∠ A O B$

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Solution

The correct option is B

$C_{1} (r, 0) C_{2} (R, 0) E q . o f A B y = r E q . o f c i r c l e C_{2} (x - R)^{2} + y^{2} = R^{2} A (R - \sqrt{R^{2} - r^{2}}, r) u s i n g R^{2} = 2 r^{2} B (R + \sqrt{R^{2} - r^{2}}, r) A (R - r, r), B (R + r, r) S l o p e o f O A = \frac{r}{R - r} = m_{1} S l o p e o f O B = \frac{r}{R + r} = m_{2} t a n θ = \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} = 1 θ = 45^{\circ}$

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Circles C1 and C2, of radii r and R respectively, touch each other as shown in figure. The line l, which is parallel to the line joining the centres of C1 and C2, is tangent to C1 at P and intersects C2 at A,B. If R2=2r2, then ∠AOB

The correct option is B C1(r,0)C2(R,0)Eq.of AB y=rEq.of circle C2(x−R)2+y2=R2A(R−√R2−r2,r) using R2=2r2B(R+√R2−r2,r) A(R−r,r),B(R+r,r)Slope of OA=rR−r=m1Slope of OB=rR+r=m2tanθ=m1−m21+m1m2=1θ=45∘

Circles $C_{1}$ and $C_{2}$ , of radii r and R respectively, touch each other as shown in figure. The line l, which is parallel to the line joining the centres of $C_{1}$ and $C_{2}$ , is tangent to $C_{1}$ at P and intersects $C_{2}$ at A,B. If $R^{2} = 2 r^{2}$ , then $∠ A O B$