TANGENT & NORMAL
1) Find the gradient/ Slope:
a) y=x²-3x+1 at (2,-1). 1
b) y= 2x²+3 sinx at x=0. 3
c) x³ - x at x=2. 11
d) x² - sinx at x=0. -1
b) y²=2x²-3x+5 at(1,2). 1/4
c) y= 3x/(x²-1). at (0,0). -3
d) log(xy)= x²- y² at (1,1). 1/3
e) y= log x at (1,0) and (e³,3). 1,1/e³
f) x = y²-4y at the point on y-axis. -1/4, 1/4
g) y=(x+1)(x-2) at (i)x-axis(ii) y-axis. -3, 3 and -1
h) calculate the gradient of the curve y= 10ˣ at the point, where it intersects the y-axis. Log 10
2) Find the Slope of the Tangent & Normal of the following:
a) y= √x³ at x= 4. 3, -1/3
b) y=√x at x=9. 1/6,-6
c) y= x³ - x at x= 2. 11, -1/11
d) y= 2x²+3sinx at x= 0. 3, -1/3
e) x= a(t - sint), y= a(1+ cost) at t= -π/2. 1, -1
f) x= a cos³t), y= a sin³t) at t= π/4. -1, 1
g) x= a(t - sint), y= a(1- cost) at t= π/2. 1, -1
h) y= (sin 2x+ cotx+2)² at t= π/2. -12, 1/12
I) x² + 3y + y²= 5 at (1,1). -2/5, 5/2
j) xy = 6 at (1,6). -6, 1/6
3) Find the values of
a) Slope of the curve xy +ax + by = 0 at (1,1) is 2. Find of a , b. 1,-2
b) Slope of the curve xy +ax + by= 3 at (1,1) is 2. Find of a , b. 6,-4
c) If the tangent to the curve y= x³+ ax + b at (1,-6) is parallel to the line x-y+5=0, a,b is . -2,-5
4) Find the point on the curve
a) y=x²-x-8 at tangent is parallel to x-axis. 1/2, -33/4
b) y= x³/3 -x²+2 tangent is parallel to x-axis. (0,2),(2,2/3)
c) y= x³- 2x² - x at which the tangent lines are parallel to the line y= 3x-2. (2,-2),(-2/3,-14/27)
d) y= x²-4x+3 at which the normal is parallel to a line whose slope is 1/2 (1,0).
e) y= x³- 3x at where the tangent is parallel to the chord joining (1,-2), (2,2). ±√(7/3),±2/3√(7/3)
f) y= x³- 2x²- 2x at which the tangent is parallel to a line y= 2x -3. (2,-4)(-2/3, 4/27)
g) y²= 2x³ at which the slope of the tangent is 3. (2,4)
h) xy+4=0 at which the tangent are inclined at an angle of 45° with the x-axis. (2,-2),(-2,2)
I) y= x² where the slope of the tangent is equal to the x-cordinates of the point. (0,0)
j) y²+x²-2x- 4y+1= 0, the tangent is parallel to x-axis. (1,0),(1,4)
k) y= x² does the tangent make an angle 45° with the x-axis. (1/2,1/4)
l) y= 3x²-9x+8 does the tangent are equally inclined with the axes. (5/3,4/3),(4/3,4/3)
m) y= 2x²-x+1 is the tangent parallel to the line y= 3x+4. (1,2)
n) y= 3x²+4 at which the tangent perpendicular to the line whose slope is -1/6. (1,7)
o) x²+ y²=13, the tangent at each one of which is parallel to the line 3y + 2x= 7. (2,3), ((-2,-3)
p) 2a²y=x³- 3ax² where the tangent is parallel x-axis. (0,0),(2a,-2a)
q) y= x²- 4x+5 is the tangent perpendicular to the line 2y+x =7. (3,2)
s) x²/4+ y²/25=1 is the tangent parallel to the
i) x-axis. (0,5),(0,-5)
ii) y-axis (2,0),(-2,0)
t) 0= x² + y² - 2x -3 is the tangent parallel to the line x-axis. (1,±2)
u) x²/9 + y²/16= 1 is the tangent are parallel to
i) x-axis. (0,-4),(0,4)
ii) y-axis. (3,0),(-3,0)
v) y= x³ where the slope of the tangent is equal to the x-cordinates of the point. (0,0),(1/3,1/27)
w) Show that the tangent to the curve y= 7x³ +11 at the point x= 2 and x= -2 are parallel.
5) Find the Equation of Tangent:
a) y²= 4ax at (0,0). x=0
b) y=x²-4x+2 at (4,2). y= 4x-14
c) y= -5x² +6x+7 at (1/2,35/4). 4x -4y+33
d) x² + 3xy+y²= 5 at (1,1). x+y= 2
e) x²+y²=25 at (3,-4). 3x-4y= 25
f) x²/a² + y²/b²= 1 at (m,n). a²ny + b²mx- a²n²- b²m²= 0.
g) ay² = x³ at (am², am³). 3mx-2y- am³= 0.
h) y= x²+ 4x+1 at the point whose abscissa is 3. 10x- y-8= 0
i) y²= x³/(4-x) at (2,-2). 2x+y-2=0
j) x²/a² - y²/b² =1 at (a sect,btant). bx sect - ay tant = ab.
k) 9x² +16y²=288 at (4,3). 3x+4y= 24
l) x² +4y² =25 at whose ordinate is 2. 3x+8y=25 & 3x-8y=25
m) y= x-sinx cosx at x =π/2. 4x-2y-π=0
n) y²= 4ax at (at², 2at). ty= x+at²
o) x= cost, y= sint, at t=π/4. x+y -√2=0
p) y= 2 sinx + sin2x at x=π/3. 2y-3√3=0
q) x=¢+sin¢, y=1+cos¢ at ¢=π/2. 2x+2y = π+4.
r) x= 1 - cost, y= t - sint at t= π/4. (√2-1)x - y +π/4 +2 - 2√2= 0
s) y= cot²x - 2 cotx +2 at x= π/4. y=1
t) √x+√y=1 at (1/4,1/4). 2x+2y-1=0
u) √x+√y=a at (a²/4,a²/4). x+y= a²/2
v) x= sin3t, y= cos2t at t=π/4. 2√2 x - 3y -2= 0
w) y= (x²-1)(x-2) at the point where the curve cuts the x-axis. -6x+y=6, 2x+y= 2, y= 3x-6.
6) Find the Equation of Normal:
a) y= 2x² + x -1 at (1,2). x+2y= 11
b) y= x² - 2x +5 at (2,5). x+2y= 12
c) y= x²+ 4x+1 at the point whose abscissa is 3. x+10y= 223
d) 9x²+ 4y²= 25 at (1,-2). 8x+ 9y+ 10 = 0
e) 4x²+ 9y²= 72 at (3,2). 3x-2y= 5
f) x²+ y²-6x- 4y+8=0 at (1,1). x-2y+1 = 0
g) x²+ y²-4x- 13=0 at (1,4). 0=4x+y-8
h) y² = 4ax at (0,0). y= 0
i) y=x²+4x+1 at x=3. x+10y = 223
j) √x +√y =1 at (1/4,1/4). y= x
k) y²= x³/(4-x) at (2,-2). 2y-x+6= 0
l) xy = c² at (ct, c/t). t³x-ty-c(t⁴-1=0
m) ³√x²+ ³√y²= ³√a² at (asin³t, acos³t). x sint - y cost - a cos 2t =0
n) x²/a² + y²/b² = 1 at (acost, b sint). ax sint - by cost = (a²-b²) sint cost.
o) x²/a² - y²/b² = 1 at (a sect, b tan t). ax tant + by sect = (a²+b²) sect tan t.
p) x= cost, y= sint, at t=π/4. y= x
q) x= 1- cost, y = t - sint at t= π/2. 2(x+y)=π
r) x= 3 cost - cos³t, y= 3 sint - sin³t at t= π/4. y= x
s) y= 2 sin² 3x at x=π/6.
7) Find the equation of the tangent line to the curve y= x²+ 4x -16 which is parallel to the line 3x-y+1= 0. 12x -4y-65= 0.
8) Find the equation of the tangent to the curve y= 2x²+ 7 which is parallel to the line 4x-y+5= 0. 4x -y+5= 0.
9) Find the equation of the tangent line to the curve y= x² - 2x +7 which is
a) parallel 2x-y+9= 0. 2x -y+3= 0
b) perpendicular to the line 5y -15x=13. 12x +36y-227= 0
10) Find the equation of the tangent line to the curve y= 2x² +7 which is parallel to the line 4x-y+3= 0. y- 4x -5= 0
11) Find the equation of the tangent line to the curve y²= 8x, which is inclined at an angle 45° with the x-axis. x - y + 3= 0.
12) Find the equation of the tangent line to the curve y= 1/(x -3), x≠ 3 with slope 2. There is no tangent to the curve that has slope 2.
13) Find the equation of the tangent line to the curve y= √(3x - 2) which is parallel to the line 4x- 2y+5= 0. 48x - 24y- 23= 0.
14) The equation of the tangent at (2,3) on the curve y²= ax³+ b is y= 4x -5. Find the values of a,b. 2,-7
15) Find the equation(s) of the tangent(s) line to the curve y= 4x³-3 x +15 which is perpendicular to the line x+9y+3= 0. 9x - y- 3= 0, 9x-y+13= 0.
16) Find the equation of the normal line to the curve y= x³+ 2x +6 which is parallel to the line x+ 14y+4= 0. x +14y+86= 0, x+14y-254= 0
17) Find the equation of the normal line to the curve y= x log x which is parallel to the line 2x- 2y+3= 0. x - y-= 3/e².
18) Find the angle of intersection
a) y²=x and x²=y. π/2, tan⁻¹3/4
b) y=x² and x²+y²=20. tan⁻¹9/2
c) 2y²= x³ & y²= 32x. π/2,tan⁻¹1/2
d) x²/a² +y²/b² =1 and x²+y²=ab. tan⁻¹{(a-b)/√(ab)}
e) x²+y²-4x-1=0, x²+y²-2y-9=0. π/4
f) x²+4y²= 8 and x²- 2y²=2. tan⁻¹3
g) x²= 27y and y²=8x. tan⁻¹9/13
h) x²+y²= 2x and y²=x. tan⁻¹1/2
I) xy= 6 and x²y=12. tan⁻¹3/11
j) y²= 4x and x²= 4y tan⁻¹3/4
k) y²= 4ax and x² =4by at their point of intersection other than the origin. tan⁻¹[3³√(ab)/{2(³√a²+ ³√b²}]
19) Show that curves intersect orthogonally:
a) y=x³ and 6y=7-x².
b) x²+4y²=8 and x²-2y²=4
c) x³-3xy²= -2 and 3x²y- y³=2.
d) y= x² and x³+6x= 7 at (1,1)
e) show that the condition that the curves ax²+by²= 1 and a'x² + b'y²=1 should intersect orthogonally is that 1/a - 1/b = 1/A' - 1/b'
20) Show that the curves intersect orthogonally at the indicated points:
a) x²=4y and 4y+x²=8 at (2,1)
b) y²=8x and 2x²+y²=10 at (1,2√2)
c) x²=y and x³ +6y=7 at (1,1)
21) Find the condition that the curves intersect orthogonally:
a)x²/a² + y²/b² =1 & xy=c². b²= a²
b) x²/a² + y²/b² =1& x²/A²-y²/B² =1. a²- b² = A²+ B².
22)
A) Show that the curves 4x=y² and 4xy=k cut at right angles if k²=512
B) Show that the curves 2x=y² and 2xy=k cut at right angles, if k²=8.
C) Show that the curves x²-3x+1=y and x(y+3)=4 cut at right angles, at the point (2,-1).
D) Show that the curves x=y² and xy=k cut at right angles, if 8k²=1
E) Show that the curves xy= a² and x² + y²= 2a² touch each other.
23) If the line x cost + y sint = p touches the parabola y² = 4ax, Prove that p= - a sint tant.
VERY SHORT ANSWER QUESTIONS
EXERCISE-- 2
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1) Find the point on the curve y= x² -2x +3, where the tangent is parallel to x-axis. . .(1,2
2) find the slope of the tangent to the curve x= t² +3t-8, y= 2t² -2t-5 at t= 2. 6/7
3) If the tangent line at a point (x,y) on the curve y= f(x) is parallel to x-axis, then write the value of dy/dx. 0
4) write the value of dy/dx, if the normal to the curve y= f(x) at (x,y) is parallel to x-axis. 0
5) if the tangent to a curve at a point (x, y) is equally inclined to the co-ordinate Axes, then write the value of dy/dx. ±1
6) If the tangent line at a point (x,y) on the curve y= f(x) is parallel to y axis, find the value of dx/dy. 0
7) find the slope of the normal at the point 't' on the curve x= 1/t, y= t. 1/t²
8) Write the coordinates of the point on the curve y²= x where the tangent line makes an angle π/4 with x-axis. (1/4,1/2)
9) Write the angle made by the tangent to the curve x= eᵗ cost, y= eᵗ sint at t=π/4 with the x-axis. π/2
10) Write the equation of the normal to the curve y= x+ sinx cosx at π/2. 2x=π
11) Find the coordinates of the point on the curve y²= 3 - 4x where tangent is parallel to the line 2x+ y-2= 0. (1/2,1)
12) write the equation of the tangent to the curve y= x² - x+2 at the point where it crosses the y-axis. x+y-2= 0
13) Write the angle between the curves y²= 4x and x²= 2y-3 at the point (1,2). 0
14) Write the angle between the curve y= e⁻ˣ and y= eˣ at their point of intersection. 90°
15) write the slope of the normal to the curve y= 1/x at the point (3,1/3). 9
16) write the coordinates of the point at which the tangent to the curve y= 2x² - x+1 is parallel to the line y= 3x+9. (1,2)
17) Write the equation of the normal to the curve y= cosx at (0,1). x= 0
MULTIPLE CHOICE QUESTIONS
1) The equation to the normal to the curve y= sinx at (0,0) is
A) x= 0 B) y= 0 C) x+y= 0 D) x-y= 0
2) The equation of the normal to the curve y= x+ sinx cosx at x=π/2 is
A) x= 2 B) x= π C) x+π= 0 D) 2x= π
3) the equation of the normal to the curve y= x(2-x) at the point (2,0) is
A) x -2y= 2 B) x -2y+2= 0
C) 2x+y= 4 D) 2x + y-4= 0
4) the point on the curve y²= x where tangent makes 45° angle with x-axis is
A)(1/2,1/4) B) (1/4,1/2)
C) (4,2) D) (1,1)
5) If the tangent to the curve x= at², y= 2at is perpendicular to x-axis, then its point of contact is
A) (a,a) B)(0,a) C)(0,0) D)(a,0)
6) the point on the curve y= x² - 3x+2 where tangent is perpendicular to y= x is
A)(0,2) B)(1,0) C) (-1,6) D)(2,-2)
7) the point on the curve y²= x where tangent makes 45° angle with x axis is
A) (1/2,1/4) B)(1/4,1/2)
C) (4,2) D) (1,1)
8) the point at the curve y= 12x - x² where the slope of the tangent is zero will be
A) (0,0) B)(2,16) C) (3,9) D) none
9) The angle between the curves y²= x and x²= y at (1,1) is
A) tan⁻ 4/3 B)tan⁻ 3/4 C) 90° D)45°
10) The equation of the normal to the curve 3x²- y²= 8 which is parallel to x+3y= 8 is
A) x+3y= 8 B) x+3y= -8
C) x+3y±8= 0 D) x+3y= 0
11) The equation of the tangent at those points where the curve y= x² - 3x+2 meets x-axis are
A) x - y= -2=x - y -1
B) x + y -2=0=x - y -2
C) x - y -1= 0=x - y
D) x - y= 0=x +y
12) the slope of the tangent to the curve x² = t²+ 3t -8, y= 2t² - 2t -5 at point (2,1) is
A) 22/7 B) 6/7 C) -6 D) none
13) At what point the slope of the tangent to the curve x²+ y²- 2x-3= 0 is zero
A)(3,0),(-1,0). B) (3,0),(1,2)
C) (-1,0),(1,2) D) (1,2),(1,-2)
14) The angle of interseption of the curve xy= a² and x² - y² = 2a² is
A) 0° B) 45° C) 90° D) none
15) If the curve ay+ x²= 7 and x³= y cut orthogonally at (1,1), then a equal to
A) 1 B) -6 C) 6 D) 0
16) If the line y= x touches the curve y= x² + bx+ c at a point (1,1) then
A) b=1, c=2 B) b=-1, c=1
C) b=2, c=1 D) b=-2, c=1
17) The slope of the tangent to the curve x= 3t²+1, y= t³ -1 at x=1 is
A) 1/2 B) 0 C) -2 D) Undefined
18) The curves y= a eˣ And y= be⁻ˣ cut orthogonally, if
A) b= a B) -b=a C) ab=1 D) ab=2
19) The Equation of the normal to the curve x= a cos³€, y= a sin³€ at the point €=π/4 is
A) x= 0 B) y= 0 C) x= y D) x+y=a
20) If the curve y= 2eˣ and y=ae⁻ˣ intersect orthogonally, then a=
A) 1/2 B) -1/2 C) 2 D) 2e²
21) The point on the curve y= 6x - x² at which the tangent to the curve is inclined at π/5 to the line x+y= 0 is
A) (-3,-27) B)(3,9) C)(7/2,35/4) D) (0,0)
22) The angle of intersection angle of the parabolas y²=4ax and x²= 4ay at the origin is
A) π/2 B) π/3 C) π/2 D)π/4
23) The angle of intersection of the curves y= 2 sin²x and y= cos2x at x=π/6 is
A) π/4 B) π/2 C) π/3 D) none
24) any tangent to the curve y= 2x⁶+ 3x+5
A) is parallel to x axis
B) is parallel to y axis
C) makes an acute angle with x-axis
D) makes an obtuse angle with x-axis
25) the point in the curve 9y²= x³, where the normal to the curve makes equal intercepts with the axes is
A)(4,8/3) B) (-4,8/3) C)(4, -8/3) D) n
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