4x​2+36y 2-4x+24y+1=0 at the point(1/2,-2/3)
Find the equations of the tangents and normals to the ellipse 

Dear student
We have, 4x2+36y2-4x+24y+1=04x2-4x+1+36y2+24y+22=222x-12+6y+22=42x-1222+6y+2222=14(2x-1)22+126y+222dydx=0dydx=-2x-136y+2So, at the point 12,-23 ,dydx12,-23=-2×12-136×-23+2=0dydx12,-23=0Thus, equation of the tangent at 12,-23 isy+23=0x-12y+23=03y+2=0And equation of the normal at 12,-23 isy+23=0x-12y+23=03y+2=0
Regards

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