Find the value of k with the straight line 3 x + 4 y equal to k touch the circle x square +y square equal to 10 x
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Toolbox:The radius of a circle is always perpendicular to the tangent.The perpendicular distance from a point (x1,y1)(x1,y1) to the line ax+by+c=0ax+by+c=0 is d=ax1+by1+ca2+b2−−−−−√d=ax1+by1+ca2+b2
Answer : k=±8k=±8
Equation of the line which touches the circle is y=3–√x+k⇒3–√x−y+k=0y=3x+k⇒3x−y+k=0
Equation of the circle is x2+y2=16x2+y2=16
⇒⇒ The coordinate of the origin is (0,0)(0,0), radius ′r′=4′r′=4.
The length of the perpendicular from (0,0)(0,0)to the line is the radius.
∴4=∣∣∣3–√(0)−(0)+k(3–√)2+(1)2−−−−−−−−−−√∣∣∴4=|3(0)−(0)+k(3)2+(1)2|
⇒4=k2⇒4=k2
∴k=±8
Answer : k=±8k=±8
Equation of the line which touches the circle is y=3–√x+k⇒3–√x−y+k=0y=3x+k⇒3x−y+k=0
Equation of the circle is x2+y2=16x2+y2=16
⇒⇒ The coordinate of the origin is (0,0)(0,0), radius ′r′=4′r′=4.
The length of the perpendicular from (0,0)(0,0)to the line is the radius.
∴4=∣∣∣3–√(0)−(0)+k(3–√)2+(1)2−−−−−−−−−−√∣∣∴4=|3(0)−(0)+k(3)2+(1)2|
⇒4=k2⇒4=k2
∴k=±8
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