Parabola.rrove

Saswati Sadual

1) Prove that the locus of the mid point of the normal chords of the parabola y²=4ax is y²/2a +4a³/y²=x-2a.

2) for what values of a will the tangents drawn to the parabola y²=4ax from a piont not on the y-axis will be normal to the parabola x²=4ay.

3) three normals to y²=4x pass through the points (15,12). show that one of the normals is given by y=x-3. also find the equation of others.

Manasi

solution to Q3:

The equation of the normal in terms of 'm' (slope) is given by,

y = mx – 2am – am³

here the parabola is y² = 4x ....i.e. a=1

therefore, the equation of the normal is y = mx – 2m – m³

now, all the 3 normals passes thru point (15,12) ...hence this point shud satisfy above equation.

substituting the value, we have 12=15m – 2m – m³

solving the equation for m, we have m = 1, 3, -4

thus the three normals are, y= x -3

y= 3x-33

y=-4x+72