COMPACT QUESTIONS TANGENT AND NORMAL

EXERCISE - A

1) Find the angle that the tangent to the hyperbola xy = c² at (c,c), makes with the x-axis.        135°

2) Find a point on the hyponova x² - 4y = 4 at which the tangent is inclined at 30° to the x-axis.      (4,√3)

3) Find the equation of the normal to the curve x³+ y³=6xy at the point (3,3).    y=x

4) The graph of y = ax + bx² passes through the point (1,0) and its gradient at the point is 1/2. Find the value of a and b.           -1/2, 1/2

5) Find the equation of the tangent to the parabola y²= 64x at the point whose ordinate is 8.        y= 4(x+1)

6) Find the point on the curve x³+ y³= 3xy where the tangent is parallel to the x-axis.    (³√2,³√2²)

7) Find the equation of the tangent to the curve y= sin x at the point x =π/6.   12y- 6√3 x = 6 -π√3.

EXERCISE - B

1) The tangent to a given curve is parallel to y -axis, if
a) dy/dx= 0 b) dy/dx= 1 c)  dx/dy= 0  d) dx/dy= 1

2) The normal to a given curve is  perpendicular to y axis, if
a) dy/dx= 0 b) dy/dx= 1 c) dx/dy= 0 d) dx/dy= 1

3) The length of the tangent from (5,1) to the circle x²+ y²+6x- 4y -3=0 is
a) 3 b) 7 c) 21 d) 49 

4) The line lx + my + n=0 is a tangent to y²=4bx, if
a) mn= bl² b) km= bn² c) ln= bm² d) bm= ln²

5) The angle that the normal to the parabola y= 2at, x= at² at t=1, makes with the x-axis is
a) 45° b) 90° c) 135° d) 180°

6) The tangent at t to the curve x= t² - 1, y= t³ - t is perpendicular to the x-axis, where 
a) t =0 b) t= 1/√3 c) t= - 1/√3  d) t= ∞



EXERCISE - C

1) Show that the line x cost + y sint =p touches the hyperbola x²/a² - y²/b² =1, if a² cos²t - b² sin²= p²

2) If the line y= 3x +5 touches the parabola y²= 8ax, find the coordinates of the point of contact.        (5/3,10)

3) Find the equation of the normal to the curve x²= 4y which passes through the point (1,2).                     x+ y= 3

4) Find the equation to the common tangents of the circle x²+ y²= 4ax and the parabola y²= 4ax.          x=0

5) Find the equation of the tangents to the hyperbola x²- 5y²= 40 which are perpendicular to the line 2x - y+3=0.      x+ 2y ±2√2=0

6) If lx+ my=1 be a normal to the hyperbola x²/a²- y²/b²= 1, prove that a²/l² - b²/m² = (a²+ b²)².         

7) Find the points on the circle x²+ y²- 6x +8y =0 at which the tangents are perpendicular to the line 4x - 3y=0.     (6,0),(0,-8)

8) Find c for which the line 2x + y= c is a normal to the curve y²= 4(x+1).      10

9) Tangents are drawn to the circle x²+ y²= a² at the points where the line x cost + ysint =p meets it. Find their point of intersection.      ((a² cost)/p, (a² sint)/p)

10) If the normal to the curve x²⁾³ + y²⁾³ = a²⁾³ makes an angle t with the x-axis, show that its equation is y cost - x sint = a cos2t.

11) Show that the normal at any point on the curve x= a(cost + t sint), y= a(sint - t cost) is at a constant distance from the origin.        









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