Identify incorrect point regarding conditions of resonance

The line&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>xsin</mi><mo>⁡</mo><mi mathvariant="normal">α</mi><mo>−</mo><mi>ycos</mi><mo>⁡</mo><mi mathvariant="normal">α</mi><mo>=</mo><mi mathvariant="normal">a</mi><mtext>&nbsp;touches the circle&nbsp;</mtext><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup><mo>=</mo><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mtext>&nbsp;then&nbsp;</mtext></math>

P is the point of intersection of lines&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>−</mo><mi>a</mi><mo>=</mo><mn>0</mn></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mi>x</mi><mo>−</mo><mi>a</mi><mi>y</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math>,&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>b</mi><mo>≠</mo><mn>0</mn></mrow></mfenced></math>.A circle with centre (1,0) and passes through P. If the tangent at P&nbsp; meets x-axes at the point&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></mfenced></math>&nbsp;then&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo></math>

If
<math >
 <m:semantics>
 <m:mrow>
 <m:mn>4</m:mn><m:msup>
 <m:mi>l</m:mi>
 <m:mn>2</m:mn>
 </m:msup>
 <m:mo>&#x2212;</m:mo><m:mn>5</m:mn><m:msup>
 <m:mi>m</m:mi>
 <m:mn>2</m:mn>
 </m:msup>
 <m:mo>+</m:mo><m:mn>6</m:mn><m:mi>l</m:mi><m:mo>+</m:mo><m:mn>1</m:mn><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow>
 </m:semantics></math>&nbsp;then the line
<math >
 <m:semantics>
 <m:mrow>
 <m:mi>l</m:mi><m:mi>x</m:mi><m:mo>+</m:mo><m:mi>m</m:mi><m:mi>y</m:mi><m:mo>+</m:mo><m:mn>1</m:mn><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow>
 </m:semantics></math>&nbsp;touches the circle ____ <pstyle="margin-top:0pt;margin-bottom:0pt;line-height:normal"><spanstyle="-aw-import:ignore">

Let the length &nbsp;of intercepts on x-axis and y-axis made by the circle&nbsp; <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup><mo>+</mo><mi>ax</mi><mo>+</mo><mn>2</mn><mi>ay</mi><mo>+</mo><mi mathvariant="normal">c</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>(</mo><mi mathvariant="normal">a</mi><mo>&lt;</mo><mn>0</mn><mo>)</mo><mtext>&nbsp;be&nbsp;</mtext><mn>2</mn><msqrt><mn>2</mn></msqrt><mtext>&nbsp;and&nbsp;</mtext><mn>2</mn><msqrt><mn>5</mn></msqrt></math>, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>=</mo><mn>0</mn></math>, is equal to

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>&nbsp;Number of values of&nbsp;</mtext><mi>c</mi><mtext>&nbsp;for which set&nbsp;</mtext><mfenced separators="|" close="}" open="{"><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>:</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>≤</mo><mn>1</mn></mrow></mfenced><mo>∩</mo><mo>{</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>:</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mi>c</mi><mo>≥</mo><mn>0</mn><mo>}</mo></math>&nbsp;contains only one element

Identify incorrect point regarding conditions of resonance

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material