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# Simultaneous Equations with a quadratic

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Prove algebraically that the straight line with equation x=2y+5 is a tangent to the circle with equation x²+y²=5.

qualitystreet  Apr 12, 2018
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Prove algebraically that the straight line with equation x=2y+5 is a tangent to the circle with equation x²+y²=5.

x=2y+5

x²+y²=5.

substitute

$$(2y+5)^2+y^2=5\\ (4y^2+20y+25)+y^2=5\\ 5y^2+20y+20=0\\ y^2+4y+4=0\\ (y+2)^2=0\\ y=-2\\ x=2*-2+5\\ x=1\\ \text{Pt of interesction is }(1,-2)$$

Since there is only one point of intersection of the line and the circle, the line must be a tangent to the circle.

Melody  Apr 12, 2018