The equation of tangent with slope m to the ellipse x2a2+y2b2=1 is given by y=mx∓√a2+b2m2.
False
Any line with slope m can be written y = mx +c. If we substitute y = mx + c in the equation x2a2+y2b2=1 we will get a quadratic. Since y = mx + c is a tangent, the quadratic should have only one solution, because tangent and the ellipse will intersect at only one point.
⇒ x2a2+(mx+c)2b2=1 has one solution or one root.
We will simplify this quadratic and say discriminant = 0
⇒ b2x2 + a2(mx+c)2 = a2b2
b2x2 + a2(m2x2+2mcx+c2)2 = a2b2
(b2 + a2m2) x2+ 2mca2x+a2c2− a2b2=0
△ = 0⇒ (2mca2)2−4(b2+a2m2)a2(c2−b2)=0
⇒ 4m2c2a4 − 4a2(b2 +a2m2)(c2 − b2) = 0
⇒ m2c2a2 − b2c2−a2m2c2+b4+b2a2m2) = 0
⇒ b2c2 + b4 + b2a2m2 = 0
⇒ −c2 + b2 + a2m2 = 0
⇒ c2 = b2 + a2m2
or c =∓√b2+a2m2