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Byju's Answer

Standard XII

Mathematics

Pair of Tangents from a Point

Circles C1 ...

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 $∠ A O B$ .

22 \frac{1^{\circ}}{2}

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45^{\circ}

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60^{\circ}

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67 \frac{1^{\circ}}{2}

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Solution

The correct option is B $45^{\circ}$
$C_{1} (r, 0)$
$C_{2} (R, 0)$
Eq. of $A B y = r$
Eq. of circle $C_{2} (x - R)^{2} + y^{2} = R^{2}$
$A (R - \sqrt{R^{2} - r^{2}}, r)$ using $R^{2} = 2 r^{2}$
$B (R + \sqrt{R^{2} - r^{2}}, r) A (R - r, r), B (R + r, r)$
Slope of $O A = \frac{r}{R - r} = m_{1}$
Slope of $O B = \frac{r}{R + r} = m_{2}$
$tan θ = \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} = 1$
$θ = 45^{\circ}$ .

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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 45∘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 $∠ A O B$ .