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a) Prove that if the line lx + my + n =0 touches the ellipse b^2 x^2 + a^2 y^2= a^2 b^2 , then a^2 l^2 +b^2 m^2 = n^2
b) If m is the gradient of the tangent from the point (3,2) to the ellipse 9x^2+16y^2=144, , find a quadratic equation in m .by nothing whether the roots of this equation are real or imaginary ,determine if the point (3,2) lies within the ellipse.
c) Find the gradient of the tangents drawn from the point (4,6) to the ellipse x^2+12y^2=48. Hence ,find the equation of the tangents and their points of contact with the ellipse.
Yes. If you write the equation of the line as and substitute that expression for y into the equation of the ellipse, then you will get a quadratic in x. That is because a typical line meets the ellipse in two points. But if the line is a tangent to the ellipse then those two points will coincide. In other words, the quadratic equation in x will have equal roots. So write down the condition for the quadratic to have equal roots. That will give you an equation connecting l, m and n, which (if you haven't made any mistakes) should look like .