# Ncert Solutions For Class 9 Maths Ex 10.1

## Ncert Solutions For Class 9 Maths Chapter 10 Ex 10.1

1. Out of the two concentric circle ,the radius of the outer circle is 5cm and the chord FC is of length 8cm is a tangent to the inner circle .Find the radius of the inner circle.

Sol.    Let the chord FC of the larger circle touch the smaller circle at the point L.

Since FC is tangent at the point L to the smaller circle with the centre O.

∴  OL$\perp$FC

Since AC is chord of the bigger circle and OL$\perp$FC.

∴ OL bisects FC

∴           FC=2FL

$\Rightarrow$            8=2FL

$\Rightarrow$          FL=4cm

Now, consider right-angled  Δ$\Delta$FLO,we obtain

OL2$OL^{2}$= FO2$FO^{2}$FL2$FL^{2}$

= 52$5^{2}$42$4^{2}$

=25-16

=9

OL =9$\sqrt{9}$=3

Hence, the radius of the smaller or inner circle is 3cm

2. Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

Sol.  Let l1$l_{1}$ and l2$l_{2}$,two intersecting lines, intersect at A, be the tangents from an external point A to a circle with centre X, at B and F respectively.

Join XB and XF

Now, in Δ$\Delta$ABX and Δ$\Delta$ARX, we have

AX=AX                        (common)

AB=AF

(tangents from an external point)

∴     Δ$\Delta$ABX=Δ$\Delta$AFX

(by SSS congruence rule)

$\Rightarrow$    $\angle$XAB=$\angle$XAF

$\Rightarrow$   X lies on the bisector of the lines l1$l_{1}$ and l2$l_{2}$.

3.In the given figure,LM and NP are common tangents to two circles of unequal radii.Prove that LM=NP.

Sol.  Let Chords LM and NP meet at the point R

Since RL and RN are tangents from an external point R to two Circles with centres O and O${O}’$.

∴      RL=RN    …..(i)

And                                            RM=RP     …..(ii)

Subtracting (ii) from (i),we have

RL-RM=RN-RP

∴        LM=NP

4.In the given figure, common tangents PQ and RS to two circles intersect at T. Prove that PQ=RS.

Sol. Clearly, TP and TR are two tangents from an external point T to the circle with centre O.

∴      TP=TR   ……(i)

Also, TQ and TS are two tangents from an external point T to the circle with centre O${O}’$.

∴      TQ=TS   …..(ii)

TP+TQ=TR+TS

$\Rightarrow$      PQ=RS

5.A chord XY of a circle is parallel to the tangent drawn at a point Z of the circle. Prove that Z bisects the arc XZY.

Sol. Since XY is parallel to the tangent drawn at the point Z and radius OZ is perpendicular to the tangent.

∴                      OR$\perp$XY

∴    OL bisects the chord XY.

∴            XL-LY

∴           arc XZ=arc ZY

i.e., Z bisects arc XZY