The number of points on the curve y=54 x⁵-135 x⁴-70 x³+180 x²+210 x at which the normal lines are parallel to x+90 y+2=0 is :

Solution

Maths sir · 4 steps

Normal slope -1/90 means tangent slope 90, so solve y' = 90 and count the real roots.

1. Slope of tangent. The line x + 90y + 2 = 0 has slope -1/90. The normal is perpendicular to the tangent, so a normal with slope -1/90 means the tangent slope is 90.
2. Set up the equation. Here y' = 270x⁴ - 540x³ - 210x² + 360x + 210. Set it equal to 90, move everything to one side and divide by 30.

   9x⁴ - 18x³ - 7x² + 12x + 4 = 0

3. Factorise. x = 2 and x = -2/3 both satisfy it, so (x-2)(3x+2) = 3x² - 4x - 4 is a factor. Dividing it out leaves 3x² - 2x - 1 = (3x+1)(x-1), so the quartic is (x-2)(x-1)(3x+2)(3x+1) = 0.
4. So. The roots x = 2, 1, -1/3, -2/3 are four distinct real values. Each one gives a separate point on the curve, so there are 4 such points, option (c).

The idea to keep. Turn a condition on the normal into a condition on the tangent: normal slope m means y' = -1/m. Then count the distinct real roots of that equation.

Answer from the NTA final answer key. Solution by PrepWiser.