Long answer questions
1.If from an external point A of a circle with centre X,two tangents AC and AD are drawn such that
Sol. Join XC and XD.
∴ [tangent to any circle is perpendicular to its radius at point of contact]
[tangents from an external point]
XA=XA [common side]
XC=XD [radii of a circle]
∴ By SSS congruency , we have
Also, AX=AC + AC
2.If x,y,z are the sides of a right triangle where c is hypotenuse,prove that the radius r of the circle which touches the sides of the triangle is given by
Sol. Let the circle touches the sides YZ,ZX,XY of the right triangle XYZ at D,E,F respectively,where YZ=x,ZX=y and XY=z.Then XE = XF and YD = YF.
3.In the given figure,from an external poiny P,a tangent PT and a line segment PXY is drawn to a circle with the centre O.ON is perpendicular on the chord XY. Prove that:
Sol. (i) PX.PY=(PN-XN)(PN+YN)
since, XN=YN,as ON
(iii)From (i) and(ii),we have
∴ In rt.
4.XY is a diameter and XZ is a chord with centre O such that
Sol. Here, XOY is a diameter of the circle, such that
[sides opp.to equal angles]
5.Two circles with centres O and
From (1) and (2),we have
From (1),we have
Hence, the required length of the common chord is 2 x PR i.e., 2×2.4 i.e., 4.8cm.