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Question: If the circle \({{x}^{2}}+{{y}^{2}}={{a}^{2}}\) cut off an intercept of length \(2l\) units from the...

If the circle x2+y2=a2{{x}^{2}}+{{y}^{2}}={{a}^{2}} cut off an intercept of length 2l2l units from the line y=mx+cy=mx+c then
A.  c2=(a2+l2)(1+m2) B. c2=(a2−l2)(1+m2) C. a2=(c2+l2)(1+m2) D. a2=(c2+l2)(1+m2) \begin{aligned} & A.\,\,{{c}^{2}}=({{a}^{2}}+{{l}^{2}})(1+{{m}^{2}}) \\\ & B.\,{{c}^{2}}=({{a}^{2}}-{{l}^{2}})(1+{{m}^{2}}) \\\ & C.\,{{a}^{2}}=({{c}^{2}}+{{l}^{2}})(1+{{m}^{2}}) \\\ & D.\,{{a}^{2}}=({{c}^{2}}+{{l}^{2}})(1+{{m}^{2}}) \\\ \end{aligned}

Explanation

Solution

In such a type of question, first compare the given circle equation with the standard equation of the circle then conclude what is the centre and radius. At last apply the formula length of the intercept cut off and calculate it.

Complete step by step solution:
We know that standard equation of circle is (x−h)2+(y−k)2=r2{{(x-h)}^{2}}+{{(y-k)}^{2}}={{r}^{2}}, where h,kh,k are the .centres and rr is the radius.
On comparing the given equation with standard form , we get
h=0  and  k=0h=0\,\,and\,\,k=0, r=ar=a
It means the equation passing through the origin centre coordinates are (h,k)=(0,0)(h,k)=(0,0) and its radius is aa.
It is also given that the length of intercept is 2l2l cut off from the line y=mx+cy=mx+c.
Formula of length of intercept =2a2−c21−m22\sqrt{{{a}^{2}}-\dfrac{{{c}^{2}}}{1-{{m}^{2}}}}
2a2−c21−m2=2l2\sqrt{{{a}^{2}}-\dfrac{{{c}^{2}}}{1-{{m}^{2}}}}=2l
22 cancel out with 22 and on squaring both sides, we get
a2−c21−m2=l2{{a}^{2}}-\dfrac{{{c}^{2}}}{1-{{m}^{2}}}={{l}^{2}}
On transferring a2{{a}^{2}} other side, we get
−c21−m2=l2−a2or−(a2−l2)-\dfrac{{{c}^{2}}}{1-{{m}^{2}}}={{l}^{2}}-{{a}^{2}}or-({{a}^{2}}-{{l}^{2}})
On rearranging the terms, we get
c2=(1+m2)(a2−l2){{c}^{2}}=(1+{{m}^{2}})({{a}^{2}}-{{l}^{2}})

Therefore option B is the correct option.

Additional information:
When a straight line passes through the circle then the straight line is called the chord of the circle.
Intercepts are the crossing points made by a circle in the axes.

Line equation- A straight line on the coordinate plane can be described by the equation
y=mx+cy=mx+c
Where mm is slope and cc is intercept, where the line crosses the y-axis.

Note:
We can solve this question by using Pythagoras theorem and distance formula. For some students Circle and its concept is difficult but if we go step by step for solution or to understand the solution, it will become easy for us.